Linear and multilinear algebra; matrix theory Proof

Proof of the singular value decomposition from the spectral theorem

Builds on Gram matrices are positive semidefinite Spectral theorem for Hermitian matrices Unitary matrix Conjugate transpose Permutation matrix One-sided inverse of a square matrix is two-sided Characteristic polynomial does not depend on the basis Singular value Orthonormal family Existence of an orthonormal basis Dimension theorem for vector spaces Dimension of the coordinate space Existence of square roots of nonnegative reals

Proof

Here $^*$ is the conjugate transpose, and $\C^k$ carries the inner product $\ip{x}{y} = y^* x$ .

A square matrix is unitary exactly when its columns form an orthonormal family. Indeed the $(i, j)$ entry of $W^* W$ is $w_i^* w_j = \ip{w_j}{w_i}$ for the columns $w_i$ of $W$ .
By the Gram matrix result, $A^* A$ is a positive semidefinite Hermitian $n \times n$ matrix.
By the spectral theorem, $A^* A = V_0 D_0 V_0^*$ with $V_0$ unitary and $D_0$ real diagonal.
Choose a permutation matrix $\Pi$ such that $D = \Pi^T D_0 \Pi$ is diagonal with entries $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_n$ . This permutes the diagonal entries of $D_0$ into decreasing order.
The matrix $\Pi$ is real with $\Pi^T \Pi = I$ , so it is unitary. By the one-sided inverse theorem, $\Pi \Pi^T = I$ . Let $V = V_0 \Pi$ . It is unitary, since $V^* V = \Pi^T V_0^* V_0 \Pi = \Pi^T \Pi = I$ .
Then $V D V^* = V_0 \Pi \Pi^T D_0 \Pi \Pi^T V_0^* = A^* A$ . Also $V V^* = I$ by the one-sided inverse theorem.
Let $v_1, \dots, v_n$ be the columns of $V$ . They are orthonormal by step 1. From $A^* A V = V D$ we get $A^* A v_i = \lambda_i v_i$ .
For each $i$ , $\lambda_i = \lambda_i v_i^* v_i = v_i^* A^* A v_i = (A v_i)^* (A v_i) \ge 0$ .
Since $D = V^* (A^* A) V$ and $V^* = V^{-1}$ , the matrices $A^* A$ and $D$ have the same characteristic polynomial. So $\lambda_1 \ge \cdots \ge \lambda_n$ are the eigenvalues of $A^* A$ , with multiplicity.
By the definition of singular values, they are $\sigma_i = \sqrt{\lambda_i}$ , using the square roots of nonnegative reals. So $\sigma_1 \ge \cdots \ge \sigma_n \ge 0$ .
Let $r$ be the number of $i$ with $\lambda_i > 0$ . Then $\lambda_i > 0$ for $i \le r$ , and $\lambda_i = 0$ for $i > r$ .
For $i \le r$ , let $u_i = A v_i / \sigma_i \in \C^m$ . By step 7, $u_i^* u_j = v_i^* A^* A v_j / (\sigma_i \sigma_j) = \lambda_j\, v_i^* v_j / (\sigma_i \sigma_j)$ .
This is $0$ for $i \ne j$ , and $\lambda_i / \sigma_i^2 = 1$ for $i = j$ . So $u_1, \dots, u_r$ is an orthonormal family in $\C^m$ .
By the existence of an orthonormal basis, it extends to an orthonormal basis $u_1, \dots, u_m$ of $\C^m$ . This basis has $m$ vectors, by the dimension theorem and the dimension of the coordinate space. So $r \le m$ .
Let $U$ be the $m \times m$ matrix with columns $u_1, \dots, u_m$ . By step 1, $U$ is unitary.
For $i > r$ , step 8 gives $(A v_i)^* (A v_i) = \lambda_i = 0$ . A vector $y$ with $y^* y = \sum_k \abs{y_k}^2 = 0$ is zero, so $A v_i = 0$ .
Let $\Sigma$ be the $m \times n$ matrix with $\Sigma_{ii} = \sigma_i$ for $i \le \min(m, n)$ and all other entries $0$ .
Column $j$ of $U \Sigma$ is $\sum_i \Sigma_{ij} u_i$ . This is $\sigma_j u_j$ if $j \le m$ , and $0$ if $j > m$ .
Column $j$ of $AV$ is $A v_j$ . For $j \le r$ , we have $j \le m$ and $A v_j = \sigma_j u_j$ by step 12.
For $j > r$ , $A v_j = 0$ by step 16. Column $j$ of $U\Sigma$ is also $0$ , since either $j > m$ or $\sigma_j = 0$ .
By steps 18 to 20, $AV = U \Sigma$ . By step 6, $A = A V V^* = U \Sigma V^*$ .
The diagonal entries of $\Sigma$ are $\sigma_1 \ge \cdots \ge \sigma_{\min(m,n)} \ge 0$ . If $m < n$ , the remaining singular values $\sigma_i$ with $i > m$ are $0$ , because $r \le m$ .
So $A = U \Sigma V^*$ with $U$ and $V$ unitary, $\Sigma_{ij} = 0$ for $i \ne j$ , and decreasing diagonal entries given by the singular values of $A$ .

Idea

The matrix $A^* A$ is Hermitian, so it has an orthonormal basis of eigenvectors $v_i$ . The images $A v_i$ are automatically orthogonal, with lengths $\sigma_i$ . Normalising them and completing to an orthonormal basis gives $U$ .

Reading path

To understand this page from the ground up, read these 653 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 that Gram matrices are positive semidefinite, 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 existence of an orthonormal basis by the Gram–Schmidt process, 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 Grassmann's formula by extending a basis of the intersection, Proof of the distributivity of matrix multiplication, Proof that the eigenvalues of a self-adjoint operator are real, Proof that the matrix of a composition is the product of the matrices, Proof that the coordinate space F^n has dimension n, 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 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 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 of the spectral theorem for self-adjoint operators by induction on the dimension, 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 that the trace is a coefficient of the characteristic polynomial by the Leibniz formula, 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 that the Gram–Schmidt vectors are orthogonal, 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 that an orthonormal family is linearly independent, 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 of the formula for the dimension of the orthogonal complement, Proof that the sign is a homomorphism, Proof of the comparison test, Proof of the existence of an eigenvalue over an algebraically closed field, 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 formula for coordinates in an orthonormal basis, 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 spectral theorem for Hermitian matrices, Proof of the product of a matrix and its adjugate by Laplace expansion, 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 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 that a space is the direct sum of a finite-dimensional subspace and its orthogonal complement, 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 that a one-sided inverse of a square matrix is two-sided, 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 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 that an intersection of subspaces is a subspace, Proof of the constant multiple rule, Proof that similar matrices have the same characteristic polynomial.

Basis concept
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Sum of subspaces definition
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Gradient concept
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Disjoint sets definition
Partial derivative definition
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Field definition
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Differentiable function of several variables definition
Integral domain definition
Associativity of matrix multiplication statement
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Distributivity of matrix multiplication statement
Union of a family of sets definition
Antiderivative definition
Differentiable function definition
Subset definition
One-sided derivative definition
Axiom of extensionality statement
Transpose of a product statement
Similar matrices definition
Unitary matrix definition
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Proof of the commutativity of addition by induction proof
Archimedean property statement
Propositional logic concept
Proof of the derivative as a little-o approximation proof
Riemann integral definition
Order on the natural numbers definition
Mathematical proof concept
A linear map is injective iff its kernel is trivial statement
The rationals form an ordered field statement
Uniqueness quantifier definition
Upper and lower Darboux sums definition
Adjoint of a linear map definition
Norm induced by an inner product definition
Orthogonal complement definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
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Partial sum definition
Intersection of subspaces statement
Orthonormal family definition
Eigenvalue definition
Spanning set definition
Alternating multilinear map definition
Direct sum of subspaces definition
Equality of complex numbers statement
Uniqueness of inverses statement
Binary relation definition
Root of a polynomial definition
Evaluation homomorphism definition
Universal set definition
Proof of the derivative of a map into R^n by components proof
Irreducible element 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
Order on the reals definition
Derivative function definition
Positive semidefinite matrix definition
Positive definite matrix definition
Determinant definition
Definite integral definition
Commutativity of addition statement
Tautology definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Sum of a finite geometric series statement
Convergent series definition
Proof that an orthonormal family is linearly independent proof
Eigenvector definition
Self-adjoint operator definition
Finite-dimensional vector space definition
Basis definition
Proof of the Steinitz exchange lemma by induction proof
Gram–Schmidt process concept
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Split polynomial definition
Sequential compactness definition
Irreducible polynomial definition
Proof of the laws of exponents in a group proof
Proof of the subgroup criterion proof
Proof that the complex numbers form a field proof
Complement definition
Interval definition
Higher derivative definition
Subspace topology definition
Proof that a matrix with two equal rows has determinant zero proof
Proof that the determinant of the identity matrix is one proof
Function definition
Characteristic polynomial definition
Minor definition
Proof that the determinant is multilinear in the rows proof
Proof that Gram matrices are positive semidefinite proof
Dimension definition
Coordinate vector definition
Dimension theorem for vector spaces statement
Strong induction statement
Proof that the eigenvalues of a self-adjoint operator are real proof
Logical equivalence definition
Sum of a series definition
Absolute convergence definition
Orthonormal basis definition
An orthonormal family is linearly independent statement
Steinitz exchange lemma statement
Proof that the Gram–Schmidt vectors are orthogonal proof
Equivalence relation definition
Preorder definition
Algebraically closed field definition
Subgroup criterion statement
Laws of exponents in a group statement
Proof of the epsilon-delta form of the limit of a function proof
Continuously differentiable function definition
Determinant of a matrix with two equal rows statement
The complex numbers form a field statement
Choice function definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the epsilon-delta form of continuity proof
Determinant of the identity matrix statement
Determinant is multilinear in the rows statement
Image of a set definition
Preimage of a set definition
Cofactor definition
Gram matrices are positive semidefinite statement
Metric space definition
Proof of the well-ordering principle from strong induction proof
Matrix of a linear map definition
Eigenvalues of a self-adjoint operator are real statement
Proof that the coordinate space F^n has dimension n proof
Proof of the formula for coordinates in an orthonormal basis proof
Proof of the double negation law by a truth table proof
Proof that a subspace of a finite-dimensional space has smaller or equal dimension proof
Gram–Schmidt vectors are orthogonal statement
Rank of a linear map definition
Nullity definition
Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
Partial order definition
Proof that disjoint cycles commute proof
Epsilon-delta definition of a limit statement
Intersection of subgroups is a subgroup statement
Left inverse definition
Right inverse definition
Axiom of choice statement
Equivalence class definition
Surjective function definition
Dimension of the coordinate space statement
Epsilon-delta definition of continuity statement
Adjugate matrix definition
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Diameter of a set definition
Coordinates in an orthonormal basis statement
Matrix of a composition is the product of matrices statement
Double negation law statement
Well-ordering principle statement
Dimension of a subspace statement
Characteristic polynomial of a linear map definition
Singular value definition
Linearly independent set of the right size is a basis statement
Maximal element definition
Total order definition
Upper bound definition
Disjoint cycles commute statement
Subgroup generated by a subset definition
Proof that equivalence classes form a partition proof
Uniform continuity definition
Bijective function definition
Composition of continuous functions is continuous 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
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Uniform continuity on a set definition
Equivalence classes form a partition statement
Open set definition
Division algorithm statement
Proof that a function is bijective iff it has a two-sided inverse proof
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
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
Open cover definition
Continuous map of topological spaces definition
Equivalent metrics definition
Zorn's lemma statement
Proof of the description of the open sets of a metric space proof
Complete metric space definition
Lipschitz functions are uniformly continuous statement
The reals form an ordered field statement
Connected space definition
Compact space definition
Proof that a set is open if and only if its complement is closed proof
Decomposition of a permutation into disjoint cycles statement
Equivalent norms definition
Open sets characterised by neighbourhoods statement
Proof that the composition of continuous maps is continuous proof
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
Connected set definition
Banach space definition
Compact set definition
Composition of continuous maps is continuous statement
Proof of the decomposition of a permutation into transpositions proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Continuity at a point of a topological space definition
Proof that closed sets contain their limits proof
Proof that open balls are open proof
Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis statement
Isomorphism with the coordinate space statement
Every linearly independent set extends to a basis statement
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals 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 of the existence of an orthonormal basis by the Gram–Schmidt process proof
Proof that the reals are Archimedean from the completeness axiom proof
Proof of Grassmann's formula by extending a basis of the intersection proof
Sign of a permutation definition
Modulus of a complex number definition
Compactness via open covers statement
Continuity via closed sets 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 uniqueness of limits proof
Proof of the squeeze theorem for sequences proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that limits preserve non-strict inequalities proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of continuity via open sets in metric spaces proof
Proof that metric spaces are Hausdorff proof
Intermediate value theorem statement
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
Rank–nullity theorem statement
Existence of an orthonormal basis statement
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Grassmann's formula statement
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that compact subsets of a Hausdorff space are closed proof
The symmetric group is generated by transpositions statement
Proof of Cantor's intersection theorem proof
Reverse triangle inequality statement
Proof that a complex number times its conjugate is the square of its modulus proof
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
Proof that an injective endomorphism of a finite-dimensional space is surjective proof
Differentiable functions are continuous statement
Proof of the sequence characterisation of uniform continuity proof
Proof of the triangle inequality for norms proof
Limit of a sum of sequences statement
Linearity of the integral statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Proof that a one-sided inverse of a square matrix is two-sided proof
Squeeze theorem statement
Proof of the monotonicity of the integral proof
Convergent sequences are bounded statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that Darboux and Riemann integrals agree proof
Proof that a space is the direct sum of a finite-dimensional subspace and its orthogonal complement 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
Sign of a cycle statement
Cantor's intersection theorem statement
Proof of the Cauchy–Riemann equations proof
Sign is a homomorphism statement
Continuous image of a compact set is compact 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
Proof that permuting rows multiplies the determinant by the sign proof
Epsilon-delta uniform continuity statement
There is a natural number n with 1/n below any positive epsilon statement
Monotonicity of the integral statement
One-sided inverse of a square matrix is two-sided statement
Darboux and Riemann integrals agree statement
Triangle inequality for norms statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
A space is the direct sum of a subspace and its orthogonal complement statement
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 of the Cauchy–Schwarz inequality proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of the formula for the dimension of the orthogonal complement proof
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
Proof of the Leibniz formula from the defining properties of the determinant proof
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Dimension of the orthogonal complement 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
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Product of continuous functions is continuous statement
Chain rule statement
Additivity of the integral over intervals statement
Operator norm definition
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Ratio test statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the Laplace expansion along a row by the Leibniz formula proof
Linearity of differentiation statement
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
Proof that the trace is a coefficient of the characteristic polynomial by the Leibniz formula 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
Laplace expansion along a row statement
Determinant is multiplicative statement
Determinant of a transpose statement
Proof of the first fundamental theorem of calculus proof
Trace is a coefficient of the characteristic polynomial 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 of the Laplace expansion along a column by transposition proof
Proof of the determinant of the inverse proof
Proof that similar matrices have the same characteristic polynomial proof
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Laplace expansion along a column statement
Determinant of the inverse statement
Continuous real function on a compact set attains its bounds statement
Characteristic polynomial does not depend on the basis statement
Proof of Fermat's theorem on stationary points proof
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 of the estimation lemma proof
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the inverse in terms of the adjugate proof
Proof of the mean value theorem for integrals 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
Chain rule in several variables statement
Proof that the integral of a continuous function is an antiderivative proof
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 existence of an eigenvalue over an algebraically closed field proof
Proof of the fundamental theorem of contour integration proof
Mean value inequality statement
Existence of an eigenvalue over an algebraically closed field 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 the spectral theorem for self-adjoint operators by induction on the dimension proof
Spectral theorem statement
Proof of the spectral theorem for Hermitian matrices proof
Spectral theorem for Hermitian matrices statement
Proof of the singular value decomposition from the spectral theorem proof

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Tag	`31AT4VJF68`
URI	`m2://31AT4VJF68`
Kind	proof
Writing	W0 stub
MSC	15A18, 15A23
Proves	Singular value decomposition
Uses	Gram matrices are positive semidefinite, Spectral theorem for Hermitian matrices, Unitary matrix, Conjugate transpose, Permutation matrix, One-sided inverse of a square matrix is two-sided, Characteristic polynomial does not depend on the basis, Singular value, Orthonormal family, Existence of an orthonormal basis, Dimension theorem for vector spaces, Dimension of the coordinate space, Existence of square roots of nonnegative reals
Changed	4 November 2026

Proof of the singular value decomposition from the spectral theorem

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