Linear and multilinear algebra; matrix theory Proof

Proof of Sylvester's criterion by induction and completing the square

Builds on Positive definite matrix Hermitian matrix Conjugate transpose Minor Fundamental theorem of algebra Eigenvalues are the roots of the characteristic polynomial Eigenvalues of a positive definite matrix are positive Determinant is the product of the eigenvalues A square matrix is invertible iff its determinant is nonzero Determinant of a triangular matrix Determinant of a block triangular matrix Determinant is multiplicative Transpose of a product

Proof

For $1 \le k \le n$ , let $A_k$ be the upper left $k \times k$ submatrix of $A$ , so that $\Delta_k = \det A_k$ is a leading minor.

Each $A_k$ is Hermitian. Indeed its $(i, j)$ entry is $a_{ij} = \overline{a_{ji}}$ , because $A$ is Hermitian.
First assume that $A$ is positive definite, and fix $k$ . Let $z \in \C^k$ with $z \ne 0$ , and let $x \in \C^n$ be $z$ followed by $n - k$ zeros.
Then $x \ne 0$ , and $x^* A x = z^* A_k z$ , where $^*$ is the conjugate transpose. So $z^* A_k z > 0$ , and $A_k$ is positive definite.
By the fundamental theorem of algebra, the characteristic polynomial of $A_k$ splits over $\C$ as $(t - \lambda_1) \cdots (t - \lambda_k)$ .
Each $\lambda_i$ is a root of the characteristic polynomial, so it is an eigenvalue of $A_k$ . By the positivity of eigenvalues and step 3, each $\lambda_i$ is a positive real.
Since the determinant is the product of the eigenvalues, $\Delta_k = \lambda_1 \cdots \lambda_k > 0$ . This proves one direction.
For the converse we argue by induction on $n$ . Assume $\Delta_k > 0$ for all $k$ . For $n = 1$ , $A = (a)$ with $a = \Delta_1 > 0$ , and $x^* A x = a \abs{x}^2 > 0$ for $x \ne 0$ .
Let $n \ge 2$ , and let $A' = A_{n-1}$ . The leading minors of $A'$ are $\Delta_1, \dots, \Delta_{n-1}$ , which are positive. By step 1 and the induction hypothesis, $A'$ is positive definite.
Since $A$ is Hermitian, it has the block form $A = \begin{pmatrix} A' & b \\ b^* & c \end{pmatrix}$ with $b \in \C^{n-1}$ and $c \in \R$ .
Since $\det A' = \Delta_{n-1} \ne 0$ , the matrix $A'$ is invertible. Conjugating $A' A'^{-1} = I$ gives $(A'^{-1})^* A' = I$ , using $(XY)^* = Y^* X^*$ from the transpose of a product.
By step 10, $(A'^{-1})^* = A'^{-1}$ , since a matrix has only one inverse.
Let $u = A'^{-1} b$ and $s = c - b^* u$ . Let $P = \begin{pmatrix} I & -u \\ 0 & 1 \end{pmatrix}$ , where $I$ is the $(n-1) \times (n-1)$ identity.
Block multiplication gives $AP = \begin{pmatrix} A' & b - A'u \\ b^* & c - b^* u \end{pmatrix} = \begin{pmatrix} A' & 0 \\ b^* & s \end{pmatrix}$ .
We have $P^* = \begin{pmatrix} I & 0 \\ -u^* & 1 \end{pmatrix}$ , and $u^* = b^* (A'^{-1})^* = b^* A'^{-1}$ by step 11. So $u^* A' = b^*$ .
By steps 13 and 14, $P^* A P = \begin{pmatrix} A' & 0 \\ b^* - u^* A' & s \end{pmatrix} = \begin{pmatrix} A' & 0 \\ 0 & s \end{pmatrix}$ .
The matrices $P$ and $P^*$ are triangular with all diagonal entries $1$ . By the determinant of a triangular matrix, $\det P = \det P^* = 1$ .
By the multiplicativity of the determinant and the determinant of a block triangular matrix, $\Delta_n = \det(P^* A P) = \Delta_{n-1}\, s$ .
So $s = \Delta_n / \Delta_{n-1} > 0$ .
Let $x \in \C^n$ with $x \ne 0$ . By step 16, $P$ is invertible. Let $y = P^{-1} x$ , so $y \ne 0$ . Write $y = (y', y_n)$ with $y' \in \C^{n-1}$ .
By step 15, $x^* A x = y^* P^* A P y = y'^* A' y' + s \abs{y_n}^2$ .
Both terms are at least $0$ , by steps 8 and 18. If $y' \ne 0$ , the first term is positive. Otherwise $y_n \ne 0$ , and the second term is positive.
So $x^* A x > 0$ for all $x \ne 0$ . As $A$ is Hermitian, $A$ is positive definite. This completes the induction and the proof.

Idea

Positive definiteness passes to the top left blocks, and a positive definite matrix has positive determinant. For the converse, a change of variables splits off the last coordinate. It writes the form as the form of $A_{n-1}$ plus $s \abs{y_n}^2$ , and the determinants show that $s = \Delta_n / \Delta_{n-1}$ is positive.

Reading path

To understand this page from the ground up, read these 617 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Steinitz exchange lemma by induction, Proof of the associativity of matrix multiplication, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that a subspace of full dimension is the whole space, Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the Laplace expansion along a column by transposition, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the fundamental theorem of algebra by Liouville's theorem, Proof that equivalence classes form a partition, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the distributivity of matrix multiplication, Proof that the constant coefficient of the characteristic polynomial is a signed determinant, Proof that the matrix of a composition is the product of the matrices, Proof of the reverse triangle inequality, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof of the derivative as a little-o approximation, Proof that disjoint cycles commute, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that scaling a row scales the determinant, Proof that a matrix is invertible iff its determinant is a unit, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof of the determinant of a block triangular matrix by the Leibniz formula, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof that a matrix with two equal rows has determinant zero, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof of the formula for the transpose of a product, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof of the inverse in terms of the adjugate, Proof of Cauchy's estimates, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that an injective endomorphism of a finite-dimensional space is surjective, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of Cauchy's integral formula for derivatives, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof that an intersection of subgroups is a subgroup, Proof of the epsilon-N definition of the limit of a sequence, Proof that a zero derivative on a connected open set implies constant, Proof of the first fundamental theorem of calculus, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the modulus is multiplicative, Proof that the sign is a homomorphism, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof of the determinant of the inverse, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that the determinant of the identity matrix is one, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that a linear map is injective iff its kernel is trivial, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof of the determinant of a scalar multiple, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof of the second fundamental theorem of calculus, Proof that the complex numbers form a field, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the determinant of a triangular matrix by the Leibniz formula, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the division algorithm, Proof of the Archimedean property in an Archimedean ordered field, Proof of the sign of a cycle, Proof that a continuous real function on a compact set attains its bounds, Proof that the eigenvalues of a positive definite matrix are positive, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a subspace of a finite-dimensional space has smaller or equal dimension, Proof that absolutely convergent series in a Banach space converge, Proof that the determinant is multilinear in the rows, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof that the eigenvalues are the roots of the characteristic polynomial, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the rank–nullity theorem by extending a basis of the kernel, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the subgroup criterion, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof that the determinant is the product of the eigenvalues by comparing constant terms, Proof of Liouville's theorem by Cauchy's estimates, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule, Proof that similar matrices have the same characteristic polynomial.

Basis concept
Binary operation definition
Complex numbers object
Complex numbers as pairs of reals concept
Composition of linear maps concept
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Cycle notation concept
Definition by recursion concept
Determinant concept
Dimension concept
Divisibility definition
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Function composition concept
Group concept
Identity matrix object
Image of a linear map concept
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Invertible matrix concept
Kernel of a linear map concept
Limit point of a set definition
Linear independence concept
Linear isomorphism concept
Linear map concept
Linear subspace concept
Lipschitz continuity definition
Local maximum definition
Mathematical induction concept
Matrix definition
Matrix multiplication concept
Matrix of a linear map concept
Multilinear map concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Nullity concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Permutation concept
Permutation definition
Proof of Zorn's lemma proof
Proposition concept
Rank of a linear map concept
Rational numbers object
Real numbers object
Scalar multiplication of matrices concept
Sequence definition
Set bounded above definition
Spanning set concept
Transposition concept
Vector space definition
p-metric on R^n example
Identity element definition
Complex exponential definition
Contour integral definition
Power series in the complex plane definition
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
A square matrix is invertible iff its determinant is nonzero statement
Group homomorphism definition
Ring definition
Subgroup definition
Symmetric group definition
Proof that the matrix of a composition is the product of the matrices proof
Ring of integers statement
Limit of a function definition
Proof that a linear map is injective iff its kernel is trivial proof
Local minimum definition
Block matrix definition
Matrix addition definition
Matrix multiplication definition
Scalar multiplication of matrices definition
Square matrix definition
Transpose definition
Successor function definition
Archimedean ordered field definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Inverse element definition
Circle integral definition
Complex conjugate definition
Disc of convergence definition
Empty set object
Family of sets definition
Ordered pair definition
Differentiable function of several variables concept
Partial derivative concept
Transposition definition
Limit of a function of several variables concept
Commutative ring definition
Parametrised curve concept
Polynomial ring definition
Ring homomorphism definition
Unit of a ring definition
Proof that an intersection of subgroups is a subgroup proof
Intersection definition
Union definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Proof of the associativity of matrix multiplication proof
Proof of the distributivity of matrix multiplication proof
Set-builder notation concept
Invertible matrix definition
Proof of the formula for the transpose of a product proof
Lower triangular matrix definition
Upper triangular matrix definition
Symmetric matrix definition
Addition of natural numbers definition
Conjugate transpose definition
Riemann integrable function definition
Biconditional definition
Truth table concept
Modus ponens concept
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Proof of the Archimedean property in an Archimedean ordered field proof
The successor function is injective statement
Zero is not a successor statement
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Inner product space definition
Linear independence definition
Composition of linear maps definition
Endomorphism definition
Image of a linear map definition
Kernel of a linear map definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
Normed vector space definition
Completeness axiom statement
Integral of 1/z around the unit circle example
Cartesian product definition
Proof of the equality of complex numbers proof
Gradient concept
Jacobian matrix concept
Proof of the uniqueness of inverses proof
Disjoint sets definition
Partial derivative definition
Differentiable function of several variables definition
Field definition
Polynomial evaluation definition
Continuous function definition
Antiderivative definition
Differentiable function definition
Union of a family of sets definition
One-sided derivative definition
Associativity of matrix multiplication statement
Distributivity of matrix multiplication statement
Axiom of extensionality statement
Subset definition
Transpose of a product statement
Similar matrices definition
Hermitian matrix definition
Proof of the commutativity of addition by induction proof
Proof of the derivative as a little-o approximation proof
Riemann integral definition
Archimedean property statement
Mathematical proof concept
Order on the natural numbers definition
Propositional logic concept
Uniqueness quantifier definition
A linear map is injective iff its kernel is trivial statement
The rationals form an ordered field statement
Upper and lower Darboux sums definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Sequential continuity definition
Norm induced by an inner product definition
Partial sum definition
Eigenvalue definition
Spanning set definition
Alternating multilinear map definition
Equality of complex numbers statement
Uniqueness of inverses statement
Binary relation definition
Proof of the derivative of a map into R^n by components proof
Evaluation homomorphism definition
Universal set definition
Partition of a set definition
Topological space definition
Derivative as a little-o approximation statement
Proof that a composition of continuous functions is continuous proof
Derivative function definition
Order on the reals definition
Definite integral definition
Positive definite matrix definition
Determinant definition
Commutativity of addition statement
Tautology definition
Upper and lower Darboux integrals definition
Proof of strong induction from ordinary induction proof
Sum of a finite geometric series statement
Convergent series definition
Finite-dimensional vector space definition
Basis definition
Proof of the Steinitz exchange lemma by induction proof
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Sequential compactness definition
Higher derivative definition
Proof of the laws of exponents in a group proof
Interval definition
Proof of the subgroup criterion proof
Proof that the complex numbers form a field proof
Complement definition
Proof that a matrix with two equal rows has determinant zero proof
Subspace topology definition
Function definition
Proof that the determinant of the identity matrix is one proof
Characteristic polynomial definition
Dimension definition
Coordinate vector definition
Dimension theorem for vector spaces statement
Minor definition
Proof that the determinant is multilinear in the rows proof
Strong induction statement
Logical equivalence definition
Proof that the eigenvalues of a positive definite matrix are positive proof
Sum of a series definition
Absolute convergence definition
Steinitz exchange lemma statement
Equivalence relation definition
Preorder definition
Continuously differentiable function definition
Subgroup criterion statement
Proof of the epsilon-delta form of the limit of a function proof
Determinant of a matrix with two equal rows statement
Laws of exponents in a group statement
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the epsilon-delta form of continuity proof
Choice function definition
The complex numbers form a field statement
Determinant of the identity matrix statement
Image of a set definition
Preimage of a set definition
Determinant is multilinear in the rows statement
Cofactor definition
Metric space definition
Eigenvalues of a positive definite matrix are positive statement
Proof of the well-ordering principle from strong induction proof
Matrix of a linear map definition
Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
Rank of a linear map definition
Nullity definition
Proof of the double negation law by a truth table proof
Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
Partial order definition
Left inverse definition
Right inverse definition
Proof that disjoint cycles commute proof
Surjective function definition
Epsilon-delta definition of a limit statement
Intersection of subgroups is a subgroup statement
Equivalence class definition
Axiom of choice statement
Epsilon-delta definition of continuity statement
Proof that scaling a row scales the determinant proof
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Adjugate matrix definition
Diameter of a set definition
Matrix of a composition is the product of matrices statement
Double negation law statement
Dimension of a subspace statement
Well-ordering principle statement
Linearly independent set of the right size is a basis statement
Characteristic polynomial of a linear map definition
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Proof that equivalence classes form a partition proof
Disjoint cycles commute statement
Uniform continuity definition
Subgroup generated by a subset definition
Composition of continuous functions is continuous statement
Scaling a row scales the determinant statement
Metric topology definition
Bounded set definition
Proof of the division algorithm proof
Proof of the diameter characterisation of Cauchy sequences proof
Proof by contradiction concept
Proof that a subspace of full dimension is the whole space proof
Algebraic multiplicity definition
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Equivalence classes form a partition statement
Uniform continuity on a set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Open set definition
Division algorithm statement
Epsilon characterisation of Cauchy sequences statement
Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
A subspace of full dimension is the whole space statement
Proof of the determinant of a scalar multiple proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Closed set definition
Proof of the decomposition of a permutation into disjoint cycles proof
Proof that open sets are the sets that are neighbourhoods of all their points proof
Continuous map of topological spaces definition
Open cover definition
Equivalent metrics definition
Proof of the description of the open sets of a metric space proof
Zorn's lemma statement
Complete metric space definition
Lipschitz functions are uniformly continuous statement
Determinant of a scalar multiple statement
The reals form an ordered field statement
Connected space definition
Proof that a set is open if and only if its complement is closed proof
Compact space definition
Open sets characterised by neighbourhoods statement
Decomposition of a permutation into disjoint cycles statement
Proof that the composition of continuous maps is continuous proof
Equivalent norms definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Continuous linear map definition
Composition of continuous maps is continuous statement
Connected set definition
Compact set definition
Banach space definition
Continuity at a point of a topological space definition
Proof of the decomposition of a permutation into transpositions proof
Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Every spanning set contains a basis statement
Isomorphism with the coordinate space statement
Every linearly independent set extends to a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Decomposition of a permutation into transpositions statement
Proof of continuity via closed sets proof
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Proof of the rank–nullity theorem by extending a basis of the kernel proof
Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Sign of a permutation definition
Modulus of a complex number definition
Continuity via closed sets statement
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof of the reverse triangle inequality proof
Proof that the symmetric group is generated by transpositions proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Intermediate value theorem statement
Proof that limits preserve non-strict inequalities proof
Proof of continuity via open sets in metric spaces proof
Proof that metric spaces are Hausdorff proof
Proof that differentiable functions are continuous proof
Rank–nullity theorem statement
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that compact subsets of a Hausdorff space are closed proof
Proof of Cantor's intersection theorem proof
Proof that a complex number times its conjugate is the square of its modulus proof
Reverse triangle inequality statement
The symmetric group is generated by transpositions statement
Proof that the sign is a homomorphism proof
Proof of the sign of a cycle proof
Lebesgue number lemma statement
Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Differentiable functions are continuous statement
Proof that an injective endomorphism of a finite-dimensional space is surjective proof
Proof of the sequence characterisation of uniform continuity proof
Limit of a sum of sequences statement
Linearity of the integral statement
Uniqueness of limits statement
Proof of the triangle inequality for norms proof
Limits preserve non-strict inequalities statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Squeeze theorem statement
Proof of the monotonicity of the integral proof
Subsequences of a convergent sequence converge to the same limit statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
Compact subsets of a Hausdorff space are closed statement
A complex number times its conjugate statement
Cantor's intersection theorem statement
Sign of a cycle statement
Proof of the Cauchy–Riemann equations proof
Continuous image of a compact set is compact statement
Sign is a homomorphism statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
An injective endomorphism of a finite-dimensional space is surjective statement
Epsilon-delta uniform continuity statement
Proof that permuting rows multiplies the determinant by the sign proof
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Entire function definition
Cauchy–Riemann equations statement
Proof that the modulus is multiplicative proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that absolutely convergent series in a Banach space converge proof
Permuting rows multiplies the determinant by the sign statement
Proof that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Decreasing sequence bounded below converges statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Limit of a product of sequences statement
Geometric series example
Proof of Riemann's criterion for integrability proof
Modulus is multiplicative statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Leibniz formula for determinants statement
Comparison test statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Bolzano–Weierstrass theorem statement
Proof of the additivity of the integral over intervals proof
Proof of the derivative of a map into a product proof
Proof of the determinant of a block triangular matrix by the Leibniz formula proof
Product of continuous functions is continuous statement
Proof that the constant coefficient of the characteristic polynomial is a signed determinant proof
Additivity of the integral over intervals statement
Chain rule statement
Ratio test statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the Heine–Borel theorem by sequential compactness proof
Linearity of differentiation statement
Operator norm definition
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof of the determinant of a triangular matrix by the Leibniz formula proof
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Proof that power series are holomorphic in the disc of convergence proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Heine–Borel theorem statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Determinant is a coefficient of the characteristic polynomial statement
Determinant of a block triangular matrix statement
Laplace expansion along a row statement
Determinant is multiplicative statement
Proof of the first fundamental theorem of calculus proof
Determinant of a transpose statement
Determinant of a triangular matrix statement
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that the determinant is the product of the eigenvalues by comparing constant terms proof
Proof of the determinant of the inverse proof
Proof of the Laplace expansion along a column by transposition proof
Proof that similar matrices have the same characteristic polynomial proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Extreme value theorem statement
Laplace expansion along a column statement
Determinant of the inverse statement
Characteristic polynomial does not depend on the basis statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Determinant is the product of the eigenvalues statement
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof of the product of a matrix and its adjugate by Laplace expansion proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Product of a matrix and its adjugate statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the estimation lemma proof
Proof of the mean value theorem for integrals proof
Proof of the inverse in terms of the adjugate proof
Linear maps on finite-dimensional spaces are continuous statement
Estimation lemma statement
Mean value theorem for integrals statement
Proof of the mean value theorem from Rolle's theorem proof
Inverse in terms of the adjugate statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that a matrix is invertible iff its determinant is a unit proof
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Proof that the integral of a continuous function is an antiderivative proof
Chain rule in several variables statement
A matrix is invertible iff its determinant is a unit statement
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof that the eigenvalues are the roots of the characteristic polynomial proof
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof of integration by substitution proof
Eigenvalues are the roots of the characteristic polynomial statement
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the mean value inequality proof
Proof of the fundamental theorem of contour integration proof
Mean value inequality statement
Fundamental theorem of contour integration statement
Proof that a zero derivative on a connected open set implies constant proof
Proof of Goursat's theorem by repeated subdivision proof
Zero derivative on a connected open set implies constant statement
Goursat's theorem statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Cauchy's integral formula statement
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof that holomorphic functions are infinitely differentiable proof
Holomorphic functions are infinitely differentiable statement
Proof of Cauchy's integral formula for derivatives proof
Cauchy's integral formula for derivatives statement
Proof of Cauchy's estimates proof
Cauchy's estimates statement
Proof of Liouville's theorem by Cauchy's estimates proof
Liouville's theorem statement
Proof of the fundamental theorem of algebra by Liouville's theorem proof
Fundamental theorem of algebra statement
Proof of Sylvester's criterion by induction and completing the square proof

Where this leads

1 page builds on this one.

Sylvester's criterion

Tag	`5BZ123RQNH`
URI	`m2://5BZ123RQNH`
Kind	proof
Writing	W0 stub
MSC	15B57, 15A15
Proves	Sylvester's criterion
Uses	Positive definite matrix, Hermitian matrix, Conjugate transpose, Minor, Fundamental theorem of algebra, Eigenvalues are the roots of the characteristic polynomial, Eigenvalues of a positive definite matrix are positive, Determinant is the product of the eigenvalues, A square matrix is invertible iff its determinant is nonzero, Determinant of a triangular matrix, Determinant of a block triangular matrix, Determinant is multiplicative, Transpose of a product
Changed	4 November 2026

Proof of Sylvester's criterion by induction and completing the square

Proof

Idea

Reading path

Where this leads

Further reading

What this page needs for W1

What links here