Linear and multilinear algebra; matrix theory Proof

Proof that the determinant of a matrix exponential is the exponential of the trace

Builds on Matrix exponential Determinant Trace of a matrix Triangularisation Upper triangular matrix Complex exponential Determinant of a triangular matrix Exponential of a sum of commuting matrices Trace of a product is symmetric Determinant is multiplicative Determinant of the identity matrix Algebra of limits

Proof

We triangularise $A$ over $\C$ and compute both sides for a triangular matrix.

If $\mathbb{K} = \R$ , we view $A$ as a complex matrix. Its exponential is given by the same series, and its determinant and trace do not change. So we may assume that $A$ is a complex matrix.
By triangularisation over $\C$ , there is an invertible complex matrix $P$ such that $U = P^{-1} A P$ is upper triangular. Write $U = (u_{ij})$ .
Let $S_N(X) = \sum_{k=0}^{N} X^k / k!$ . By induction on $k$ , $U^k = P^{-1} A^k P$ . So $S_N(U) = P^{-1} S_N(A) P$ .
Each entry of $P^{-1} S_N(A) P$ is a fixed linear combination of the entries of $S_N(A)$ . By the definition of $e^A$ and the algebra of limits, letting $N \to \infty$ in step 3 gives $e^U = P^{-1} e^A P$ .
Let $X, Y$ be upper triangular. For $i > j$ , each term $x_{il} y_{lj}$ of $(X Y)_{ij}$ needs $i \le l \le j$ , which is impossible. So $X Y$ is upper triangular. The only nonzero term of $(X Y)_{ii}$ has $l = i$ , so $(X Y)_{ii} = x_{ii} y_{ii}$ .
By step 5 and induction on $k$ , $U^k$ is upper triangular with diagonal entries $u_{ii}^k$ . This also holds for $k = 0$ , with $U^0 = I_n$ .
So $S_N(U)$ is upper triangular, and its entry $(i, i)$ is $\sum_{k=0}^{N} u_{ii}^k / k!$ .
Letting $N \to \infty$ , $e^U$ is upper triangular. Its entry $(i, i)$ is the complex exponential $e^{u_{ii}}$ .
By the determinant of a triangular matrix, $\det e^U = e^{u_{11}} e^{u_{22}} \cdots e^{u_{nn}}$ .
For complex numbers $z, w$ , the $1 \times 1$ matrices $(z)$ and $(w)$ commute, and their exponentials are $(e^z)$ and $(e^w)$ . By the exponential of a sum of commuting matrices, $e^{z + w} = e^z e^w$ .
By step 10 and induction on $n$ , $e^{u_{11}} \cdots e^{u_{nn}} = e^{u_{11} + \cdots + u_{nn}} = e^{\tr U}$ , where $\tr$ is the trace.
Since the trace of a product is symmetric, $\tr U = \tr\big(P^{-1} (A P)\big) = \tr\big((A P) P^{-1}\big) = \tr A$ .
By step 4, $e^A = P e^U P^{-1}$ . The determinant is multiplicative, and complex numbers commute. So $\det e^A = \det e^U \cdot \det P \det P^{-1} = \det e^U \det(P P^{-1})$ .
Since $P P^{-1} = I_n$ and $\det I_n = 1$ , step 13 gives $\det e^A = \det e^U$ .
By steps 9, 11, 12 and 14, $\det e^A = e^{\tr U} = e^{\tr A}$ .

Idea

Both sides are unchanged when $A$ is replaced by a similar matrix. Over $\C$ , every matrix is similar to a triangular one. For a triangular matrix $U$ , the exponential is triangular with diagonal entries $e^{u_{ii}}$ , so its determinant is $\prod_i e^{u_{ii}} = e^{\tr U}$ .

Reading path

To understand this page from the ground up, read these 708 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 Cauchy–Hadamard theorem, Proof of the associativity of matrix multiplication, Proof of the Weierstrass M-test, 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 formula for the derivative of a polynomial, 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 term-by-term differentiation, 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 power rule by induction, Proof that absolutely convergent series converge, 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 of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, 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 that the exponential series has infinite radius of convergence, 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 the root test, 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 that power series converge uniformly on smaller intervals inside the radius of convergence, 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 power series are differentiable inside the radius of convergence, 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 uniform convergence of derivatives allows term-by-term differentiation, 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 Pascal's rule from the recursive definition, 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 that Cauchy sequences are bounded, 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 that the trace of a product is symmetric, 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 the factorial formula for binomial coefficients by induction, 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 of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, Proof of triangularisation by induction on the size, 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 that the exponential of a sum of commuting matrices is the product of the exponentials, 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 the exponential of a sum by the Cauchy product, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof of the power series of the exponential function, Proof that convergent sequences are bounded, Proof of the product rule, 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 of the ratio formula for the radius of convergence, 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 the Cauchy product of absolutely convergent series converges absolutely to the product, 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.

Basis concept
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Change of basis matrix concept
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Integral of 1/z around the unit circle example
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Gradient concept
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Proof of the equality of complex numbers proof
Divisibility in a commutative ring definition
Proof of the uniqueness of inverses proof
Degree of a polynomial definition
Polynomial evaluation definition
Field definition
Disjoint sets definition
Integral domain definition
Partial derivative definition
Differentiable function of several variables definition
Continuous function definition
Antiderivative definition
Differentiable function definition
One-sided derivative definition
Associativity of matrix multiplication statement
Distributivity of matrix multiplication statement
Union of a family of sets definition
Subset definition
Axiom of extensionality statement
Proof that the trace of a product is symmetric proof
Similar matrices definition
Proof of the commutativity of addition by induction proof
Proof of the derivative as a little-o approximation proof
Archimedean property statement
Riemann integral definition
Propositional logic concept
Mathematical proof concept
Order on the natural numbers definition
A linear map is injective iff its kernel is trivial statement
Uniqueness quantifier definition
Limit superior definition
The rationals form an ordered field statement
Proof of the sum of a finite geometric series proof
Upper and lower Darboux sums definition
Algebra of limits concept
Convergent sequence definition
Pointwise convergence definition
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The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Norm induced by an inner product definition
Partial sum definition
Power series definition
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Spanning set definition
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Equality of complex numbers statement
Factorial formula for binomial coefficients statement
Leading coefficient definition
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Uniqueness of inverses statement
Evaluation homomorphism definition
Proof of cancellation in an integral domain proof
Binary relation definition
Irreducible element definition
Universal set definition
Proof of the derivative of a map into R^n by components proof
Proof that a composition of continuous functions is continuous proof
Derivative as a little-o approximation statement
Topological space definition
Derivative function definition
Partition of a set definition
Order on the reals definition
Trace of a product is symmetric statement
Definite integral definition
Commutativity of addition statement
Determinant definition
Tautology definition
Proof of strong induction from ordinary induction proof
Upper and lower Darboux integrals definition
Sum of a finite geometric series statement
Uniform convergence definition
Radius of convergence definition
Convergent series definition
Eigenvector definition
Finite-dimensional vector space definition
Basis definition
Proof of the Steinitz exchange lemma by induction proof
Monic polynomial definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Split polynomial definition
Symmetric relation definition
Transitive relation definition
Proof of the degree of a product over an integral domain proof
Proof of the remainder theorem proof
Cancellation in an integral domain statement
Irreducible polynomial definition
Proof that the complex numbers form a field proof
Proof of the laws of exponents in a group proof
Proof of the subgroup criterion proof
Sequential compactness definition
Higher derivative definition
Complement definition
Interval definition
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Proof that a matrix with two equal rows has determinant zero proof
Function definition
Proof that the determinant of the identity matrix is one proof
Characteristic polynomial definition
Uniform convergence on a set definition
Minor definition
Dimension definition
Coordinate vector definition
Dimension theorem for vector spaces statement
Strong induction statement
Proof that the determinant is multilinear in the rows proof
Logical equivalence definition
Sum of a series definition
Absolute convergence definition
Series of functions definition
Steinitz exchange lemma statement
Binomial theorem statement
Preorder definition
Equivalence relation definition
Division algorithm for polynomials statement
Algebraically closed field definition
Degree of a product statement
Subgroup criterion statement
Taylor polynomial definition
Continuously differentiable function definition
Laws of exponents in a group statement
Proof of the epsilon-delta form of the limit of a function proof
The complex numbers form a field statement
Proof of the epsilon-delta form of continuity proof
Determinant of a matrix with two equal rows statement
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Choice function definition
Determinant of the identity matrix statement
Image of a set definition
Preimage of a set definition
Cofactor definition
Determinant is multilinear in the rows statement
Metric space definition
Change of basis matrix definition
Proof of the well-ordering principle from strong induction proof
Matrix of a linear map definition
Proof that the coordinate space F^n has dimension n 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
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
Remainder theorem statement
Smooth function definition
Proof that R[x] is an integral domain iff R is proof
Proof that disjoint cycles commute proof
Epsilon-delta definition of a limit statement
Intersection of subgroups is a subgroup statement
Equivalence class definition
Epsilon-delta definition of continuity statement
Left inverse definition
Right inverse definition
Surjective function definition
Axiom of choice statement
Dimension of the coordinate space 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
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
Linearly independent set of the right size is a basis statement
Maximal element definition
Total order definition
Upper bound definition
Proof of the factor theorem proof
R[x] is an integral domain iff R is statement
Uniform continuity definition
Disjoint cycles commute statement
Subgroup generated by a subset definition
Composition of continuous functions is continuous statement
Proof that equivalence classes form a partition proof
Bijective function definition
Metric topology definition
Taylor series 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 every sequence has a monotone subsequence, by peaks proof
Proof that a subspace of full dimension is the whole space proof
Chain definition
Factor theorem statement
Uniform continuity on a set definition
Equivalence classes form a partition statement
Maclaurin series definition
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
Proof of the decomposition of a permutation into disjoint cycles proof
Closed set definition
Continuous map of topological spaces definition
Open cover definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Zorn's lemma statement
Equivalent metrics definition
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
Connected space definition
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
Continuous linear map definition
Banach space definition
Composition of continuous maps is continuous statement
Compact set definition
Connected set definition
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the decomposition of a permutation into transpositions 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
Isomorphism with the coordinate space statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Triangle inequality statement
Absolute value is multiplicative 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
Closed sets contain their limits statement
Proof of continuity via closed sets proof
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the rank–nullity theorem by extending a basis of the kernel proof
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
Proof of the limit of a sum of sequences proof
Modulus of a complex number definition
Continuity via closed sets statement
Compactness via open covers statement
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that the symmetric group is generated by transpositions proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof of the Lebesgue number lemma by a finite cover of balls 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 sequence 1/n tends to zero example
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Proof that Cauchy sequences are bounded proof
Proof of the term test 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
Proof that a complex number times its conjugate is the square of its modulus proof
The symmetric group is generated by transpositions statement
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
Epsilon-delta continuity in metric spaces statement
Metric spaces are Hausdorff statement
Proof that compact subsets of a metric space are bounded proof
Differentiable functions are continuous 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
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the sequence characterisation of uniform continuity proof
Linearity of the integral statement
Proof of the triangle inequality for norms proof
Proof of the ratio formula for the radius of convergence 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
Subsequences of a convergent sequence converge to the same limit statement
Proof of the monotonicity of the integral proof
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
Cauchy sequences are bounded 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
Sign is a homomorphism statement
Continuous image of a compact set is compact statement
Proof of the Cauchy–Riemann equations proof
Proof of the limit of a product of sequences proof
Continuity via open sets statement
Compactness and sequential compactness agree in metric spaces 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
One-sided inverse of a square matrix is two-sided statement
Term test statement
Darboux and Riemann integrals agree statement
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
Proof that a series of nonnegative terms converges iff its partial sums are bounded 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 of the sequential characterization of limits proof
Proof that Lipschitz maps are uniformly continuous 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
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Geometric series example
Proof of Riemann's criterion for integrability proof
A series of nonnegative terms converges iff its partial sums are bounded statement
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
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the root test proof
Proof of the ratio test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Leibniz formula for determinants statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Comparison test statement
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Proof of the additivity of the integral over intervals proof
Exponentials dominate powers in sequences statement
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
Product rule statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Additivity of the integral over intervals statement
Operator norm definition
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the Laplace expansion along a row by the Leibniz formula proof
Ratio test statement
Proof of the determinant of a triangular matrix by the Leibniz formula proof
Proof that the determinant is multiplicative by expanding the rows proof
Root test statement
Proof that the determinant of a transpose equals the determinant proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Derivative of a map into a product statement
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof that power series are holomorphic in the disc of convergence proof
Heine–Borel theorem statement
Proof of the power rule by induction proof
Boundedness theorem statement
Determinant of a block triangular matrix statement
Laplace expansion along a row statement
Proof of the first fundamental theorem of calculus proof
Determinant is multiplicative statement
Ratio formula for the radius of convergence statement
Determinant of a transpose statement
Cauchy–Hadamard theorem statement
Determinant of a triangular matrix statement
Cauchy criterion for sequences 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
Power rule statement
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
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Determinant of the inverse statement
Laplace expansion along a column statement
Characteristic polynomial does not depend on the basis statement
Cauchy criterion for series statement
Derivative of a polynomial 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 that absolutely convergent series converge 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
Absolutely convergent series converge 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
Proof of the Weierstrass M-test 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
Weierstrass M-test 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 power series converge uniformly on smaller intervals inside the radius of convergence 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
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
A matrix is invertible iff its determinant is a unit statement
Power series converge uniformly on smaller discs inside the radius of convergence 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
Uniform convergence of derivatives allows term-by-term differentiation 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 that the exponential series has infinite radius of convergence proof
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof of integration by substitution proof
The exponential series has infinite radius of convergence statement
Proof of term-by-term differentiation 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
Term-by-term differentiation statement
Exponential function definition
Proof of the fundamental theorem of contour integration proof
Proof of the exponential of a sum by the Cauchy product proof
Mean value inequality statement
Proof that power series are differentiable inside the radius of convergence proof
Fundamental theorem of contour integration statement
Exponential of a sum statement
Proof that a zero derivative on a connected open set implies constant proof
Power series are differentiable inside the radius of convergence statement
Proof of Goursat's theorem by repeated subdivision proof
Zero derivative on a connected open set implies constant statement
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
Goursat's theorem statement
The exponential function is its own derivative statement
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Proof of the power series of the exponential function proof
Cauchy's integral formula statement
Power series of the exponential function statement
Proof that holomorphic functions are analytic proof
Proof that the exponential of a sum of commuting matrices is the product of the exponentials proof
Holomorphic functions are analytic statement
Exponential of a sum of commuting matrices 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
Triangularisation statement
Proof that the determinant of a matrix exponential is the exponential of the trace proof

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Determinant of a matrix exponential

Tag	`W69VC0G18A`
URI	`m2://W69VC0G18A`
Kind	proof
Writing	W0 stub
MSC	15A16, 15A15
Proves	Determinant of a matrix exponential
Uses	Matrix exponential, Determinant, Trace of a matrix, Triangularisation, Upper triangular matrix, Complex exponential, Determinant of a triangular matrix, Exponential of a sum of commuting matrices, Trace of a product is symmetric, Determinant is multiplicative, Determinant of the identity matrix, Algebra of limits
Changed	4 November 2026

Proof that the determinant of a matrix exponential is the exponential of the trace

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