Linear and multilinear algebra; matrix theory Proof

Proof of triangularisation by induction on the size

Builds on Characteristic polynomial Upper triangular matrix Similar matrices Characteristic polynomial does not depend on the basis Determinant of a triangular matrix Eigenvalues are the roots of the characteristic polynomial Eigenvector Every linearly independent set extends to a basis Change of basis matrix Change of basis formula Matrix of a linear map Determinant of a block triangular matrix Cancellation in an integral domain R[x] is an integral domain iff R is Block diagonal matrix Factor theorem Algebraically closed field Fundamental theorem of algebra

Proof

The forward implication is a determinant computation. The converse is by induction on $n$ .

Suppose $A$ is similar to an upper triangular matrix $U = (u_{ij})$ . Similar matrices have the same characteristic polynomial, so $p_A = p_U$ .
The matrix $t I_n - U$ is upper triangular over the commutative ring $F[t]$ , with diagonal entries $t - u_{ii}$ . By the determinant of a triangular matrix, $p_U = (t - u_{11}) \cdots (t - u_{nn})$ . So $p_A$ splits over $F$ .
For the converse, we prove by induction on $n \ge 1$ : if $p_A$ splits over $F$ , then $A$ is similar to an upper triangular matrix. For $n = 1$ , every $1 \times 1$ matrix is upper triangular.
Let $n \ge 2$ , and assume the claim for size $n - 1$ . The characteristic polynomial $p_A$ is monic of degree $n$ . As it splits, $p_A = \prod_{i=1}^{s} (a_i t + b_i)$ with $a_i \ne 0$ . Comparing degrees gives $s = n$ , and comparing leading coefficients gives $\prod_i a_i = 1$ . With $\lambda_i = -b_i / a_i$ , we get $p_A = (t - \lambda_1) \cdots (t - \lambda_n)$ .
Let $L \colon F^n \to F^n$ be $L(x) = A x$ . Its matrix in the standard basis is $A$ , so $p_L = p_A$ . Since $p_L(\lambda_1) = 0$ , the scalar $\lambda_1$ is an eigenvalue of $L$ by the roots of the characteristic polynomial. Choose an eigenvector $v$ , so $v \ne 0$ and $A v = \lambda_1 v$ .
The set $\set{v}$ is linearly independent, since $v \ne 0$ . By extension to a basis, there is a basis $B' = (v, w_2, \dots, w_n)$ of $F^n$ .
Let $Q$ be the matrix with columns $v, w_2, \dots, w_n$ . It is the change of basis matrix from the standard basis to $B'$ . The matrix of $L$ in the standard basis is $A$ .
By the change of basis formula, $Q$ is invertible and $Q^{-1} A Q = [L]_{B'}$ . Since $L(v) = \lambda_1 v$ , the first column of $[L]_{B'}$ is $\lambda_1 e_1$ .
So $Q^{-1} A Q = \begin{pmatrix} \lambda_1 & r \\ 0 & A' \end{pmatrix}$ for some $1 \times (n-1)$ row $r$ and some $(n-1) \times (n-1)$ matrix $A'$ .
By step 9 and the determinant of a block triangular matrix over $F[t]$ , $p_{Q^{-1} A Q} = (t - \lambda_1) p_{A'}$ . By invariance under similarity, this equals $p_A$ .
So $(t - \lambda_1) p_{A'} = (t - \lambda_1) \prod_{i=2}^{n} (t - \lambda_i)$ . The ring $F[t]$ is an integral domain, so cancellation gives $p_{A'} = \prod_{i=2}^{n} (t - \lambda_i)$ . Thus $p_{A'}$ splits.
By the induction hypothesis, there is an invertible $(n-1) \times (n-1)$ matrix $R$ such that $U' = R^{-1} A' R$ is upper triangular.
Let $S$ be the block diagonal matrix $\operatorname{diag}(1, R)$ . It is invertible, with inverse $\operatorname{diag}(1, R^{-1})$ .
Block multiplication gives $S^{-1} Q^{-1} A Q S = \begin{pmatrix} \lambda_1 & r R \\ 0 & U' \end{pmatrix}$ . This matrix is upper triangular, since $U'$ is.
With the invertible matrix $P = Q S$ , step 14 says $P^{-1} A P$ is upper triangular. This completes the induction.
Now let $F$ be algebraically closed. We show by induction on $d \ge 1$ that each monic $f \in F[t]$ of degree $d$ splits. The polynomial $f$ has a root $c \in F$ , and the factor theorem gives $f = (t - c) g$ .
Here $g$ is monic of degree $d - 1$ . If $d = 1$ , then $g = 1$ . If $d \ge 2$ , then $g$ splits by the induction hypothesis, so $f$ splits.
So over an algebraically closed field, $p_A$ splits, and $A$ is similar to an upper triangular matrix by step 15.
By the fundamental theorem of algebra, each nonconstant complex polynomial has a complex root. So $\C$ is algebraically closed, and step 18 applies to $\C$ .

Idea

A triangular matrix visibly has a split characteristic polynomial. Conversely, a root of $p_A$ gives an eigenvector. Putting it first in a basis makes the first column $\lambda_1 e_1$ . The remaining $(n-1) \times (n-1)$ block still has a split characteristic polynomial, so induction triangularises it.

Reading path

To understand this page from the ground up, read these 630 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 cancellation in an integral domain, 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 change of basis formula, Proof of the distributivity of matrix multiplication, 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 that R[x] is an integral domain iff R is, 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 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 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 of the degree of a product over an integral domain, 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 of the remainder theorem, 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 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 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 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 of the factor theorem, 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 of the constant multiple rule, Proof that similar matrices have the same characteristic polynomial.

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Function composition definition
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Metric space definition
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Linearly independent set of the right size is a basis statement
Characteristic polynomial of a linear map definition
Maximal element definition
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Proof of the factor theorem proof
R[x] is an integral domain iff R is statement
Uniform continuity definition
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Proof that equivalence classes form a partition proof
Composition of continuous functions is continuous statement
Bijective function definition
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Proof by contradiction concept
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Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Factor theorem statement
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Proof that a function is bijective iff it has a two-sided inverse proof
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Proof that Lipschitz functions are uniformly continuous proof
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Closed set definition
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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
Compact space definition
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Decomposition of a permutation into disjoint cycles statement
Proof that a set is open if and only if its complement is closed proof
Equivalent norms definition
Open sets characterised by neighbourhoods statement
Proof that the composition of continuous maps is continuous proof
Absolute value definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
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
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Compact set definition
Composition of continuous maps is continuous statement
Proof of the decomposition of a permutation into transpositions proof
Connected set definition
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
Every linearly independent set extends to a basis statement
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Existence of the infimum statement
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Proof that compactness of a subset can be tested with open sets of the ambient space proof
Triangle inequality statement
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Proof of continuity via closed sets proof
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
Proof of the reverse triangle inequality proof
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Compactness via open covers statement
Continuity via closed sets statement
Proof of the squeeze theorem for sequences proof
Proof that the symmetric group is generated by transpositions proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof that metric spaces are Hausdorff proof
Proof of continuity via open sets in metric spaces proof
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Rank–nullity theorem statement
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
Reverse triangle inequality statement
Complex differentiability definition
Proof that compact subsets of a Hausdorff space are closed proof
The symmetric group is generated by transpositions statement
Proof that a complex number times its conjugate is the square of its modulus proof
Proof of Cantor's intersection theorem proof
Proof that the sign is a homomorphism proof
Lebesgue number lemma statement
Proof of the sign of a cycle proof
Proof that a continuous image of a compact set is compact proof
Proof that compact subsets of a metric space are bounded proof
Epsilon-delta continuity in metric spaces statement
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
Limits preserve non-strict inequalities statement
Proof of the triangle inequality for norms proof
Squeeze theorem statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Proof of the monotonicity of the integral proof
Subsequences of a convergent sequence converge to the same limit statement
Proof that a one-sided inverse of a square matrix is two-sided proof
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
Sign of a cycle statement
A complex number times its conjugate statement
Cantor's intersection theorem statement
Sign is a homomorphism statement
Proof of the Cauchy–Riemann equations proof
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Proof of the limit of a product of sequences proof
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
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
One-sided inverse of a square matrix is two-sided 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 of the change of basis formula 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
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
Change of basis formula 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
Comparison test statement
Leibniz formula for determinants 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
Chain rule statement
Additivity of the integral over intervals statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Ratio test statement
Linearity of differentiation statement
Operator norm definition
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the determinant of a triangular matrix by the Leibniz formula proof
Proof of the Laplace expansion along a row 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
Proof of the constant multiple rule proof
Heine–Borel theorem statement
Boundedness theorem statement
Determinant of a block triangular matrix statement
Proof of the first fundamental theorem of calculus proof
Determinant is multiplicative statement
Laplace expansion along a row statement
Determinant of a transpose statement
Determinant of a triangular matrix statement
Power series are holomorphic in the disc of convergence statement
Constant multiple rule statement
Derivative of a map into R^n by components statement
Proof of the determinant of the inverse proof
Proof that similar matrices have the same characteristic polynomial proof
Proof of the Laplace expansion along a column by transposition 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
Determinant of the inverse statement
Continuous real function on a compact set attains its bounds statement
Laplace expansion along a column statement
Proof of Fermat's theorem on stationary points proof
Characteristic polynomial does not depend on the basis 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
Equivalence of norms in finite dimension statement
Product of a matrix and its adjugate 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 triangularisation by induction on the size proof

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Tag	`PQ79GJDGRQ`
URI	`m2://PQ79GJDGRQ`
Kind	proof
Writing	W0 stub
MSC	15A20, 15A21
Proves	Triangularisation
Uses	Characteristic polynomial, Upper triangular matrix, Similar matrices, Characteristic polynomial does not depend on the basis, Determinant of a triangular matrix, Eigenvalues are the roots of the characteristic polynomial, Eigenvector, Every linearly independent set extends to a basis, Change of basis matrix, Change of basis formula, Matrix of a linear map, Determinant of a block triangular matrix, Cancellation in an integral domain, R[x] is an integral domain iff R is, Block diagonal matrix, Factor theorem, Algebraically closed field, Fundamental theorem of algebra
Changed	4 November 2026

Proof of triangularisation by induction on the size

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