Proof of the spectral theorem for self-adjoint operators by induction on the dimension

Builds on Inner product Self-adjoint operator Eigenvalue Eigenvector Orthonormal basis Existence of an orthonormal basis Coordinates in an orthonormal basis Existence of an eigenvalue over an algebraically closed field Fundamental theorem of algebra Algebraically closed field Eigenvalues of a self-adjoint operator are real Orthogonal complement Dimension of the orthogonal complement An orthonormal family is linearly independent Linearly independent set of the right size is a basis

Proof

We argue by induction on $n = \dim V$ . Steps 2 to 11 show that $T$ has an eigenvalue in the scalar field.

Let $\mathbb{K}$ be $\R$ or $\C$ . For $n \ge 0$ , let $P(n)$ be the following claim. Let $V$ be an $n$ -dimensional inner product space over $\mathbb{K}$ , and let $T \colon V \to V$ be self-adjoint. Then $V$ has an orthonormal basis of eigenvectors of $T$ .
Base case $n = 0$ . Then $V = \set{0}$ , and the empty family is an orthonormal basis. It vacuously consists of eigenvectors. So $P(0)$ holds.
Let $n \ge 1$ , and let $V$ and $T$ be as in $P(n)$ , with inner product $\ip{\cdot}{\cdot}$ . If $\mathbb{K} = \C$ , then $\C$ is algebraically closed by the fundamental theorem of algebra. So $T$ has an eigenvalue.
Now let $\mathbb{K} = \R$ . Take an orthonormal basis $e_1, \dots, e_n$ of $V$ , and set $a_{ij} = \ip{T(e_j)}{e_i} \in \R$ . By the formula for coordinates, $T(e_j) = \sum_{i} a_{ij} e_i$ .
Over $\R$ , conjugate symmetry is symmetry. So self-adjointness gives $a_{ji} = \ip{T(e_i)}{e_j} = \ip{e_i}{T(e_j)} = \ip{T(e_j)}{e_i} = a_{ij}$ . Let $A = (a_{ij})$ , a real symmetric matrix.
View $A$ as a complex matrix. Since $\C^n \ne \set{0}$ , the existence of an eigenvalue over $\C$ , applied to $z \mapsto Az$ , gives $\lambda \in \C$ and $z \in \C^n$ with $z \ne 0$ and $Az = \lambda z$ .
Set $s = \sum_{i,j} \overline{z_i}\, a_{ij} z_j$ . Since $\sum_j a_{ij} z_j = \lambda z_i$ , we get $s = \lambda \sum_i \abs{z_i}^2$ .
Since the $a_{ij}$ are real, $\overline{s} = \sum_{i,j} z_i a_{ij} \overline{z_j}$ . Exchanging the names $i$ and $j$ and using $a_{ji} = a_{ij}$ gives $\overline{s} = s$ . So $s$ is real.
The number $\sum_i \abs{z_i}^2$ is real and positive, as $z \ne 0$ . By step 7, $\lambda$ is real.
Write $z = x + i y$ with $x, y \in \R^n$ . Since $A$ and $\lambda$ are real, comparing real and imaginary parts of $Az = \lambda z$ gives $Ax = \lambda x$ and $Ay = \lambda y$ .
As $z \ne 0$ , one of $x, y$ is nonzero; call it $c$ . Set $v = \sum_j c_j e_j$ , which is nonzero. By step 4, $T(v) = \sum_i \big(\sum_j a_{ij} c_j\big) e_i = \sum_i \lambda c_i e_i = \lambda v$ . So $\lambda \in \R$ is an eigenvalue of $T$ .
In both cases, let $\lambda$ be an eigenvalue of $T$ and $v$ an eigenvector for it. By the theorem on eigenvalues, $\lambda$ is real.
Set $e = v / \norm{v}$ . Then $\ip{e}{e} = 1$ and $T(e) = \lambda e$ .
Let $W$ be the orthogonal complement of the line $L = \set{a e : a \in \mathbb{K}}$ . A vector $w$ lies in $W$ if and only if $\ip{w}{e} = 0$ , by linearity in the first argument and conjugate symmetry.
By the dimension formula, $\dim W = n - \dim L = n - 1$ .
Let $w \in W$ . Then $\ip{T(w)}{e} = \ip{w}{T(e)} = \ip{w}{\lambda e} = \overline{\lambda \ip{e}{w}} = \lambda \overline{\ip{e}{w}} = \lambda \ip{w}{e} = 0$ , as $\lambda$ is real. So $T(w) \in W$ .
The restriction of $\ip{\cdot}{\cdot}$ to $W$ is an inner product. Let $S \colon W \to W$ be $S(w) = T(w)$ . It maps into $W$ by step 16, and it is linear since $T$ is.
For $w, w' \in W$ , $\ip{S(w)}{w'} = \ip{T(w)}{w'} = \ip{w}{T(w')} = \ip{w}{S(w')}$ . So $S$ is self-adjoint on $W$ .
By $P(n - 1)$ , $W$ has an orthonormal basis $f_2, \dots, f_n$ of eigenvectors of $S$ . Each $f_k$ is nonzero and satisfies $T(f_k) = S(f_k) = \mu_k f_k$ for some $\mu_k$ , so it is an eigenvector of $T$ .
Put $f_1 = e$ . Each $f_k$ with $k \ge 2$ lies in $W$ , so $\ip{f_k}{f_1} = 0$ , and $\ip{f_1}{f_k} = \overline{0} = 0$ . With steps 13 and 19, $f_1, \dots, f_n$ is orthonormal.
An orthonormal family is linearly independent. These $n$ vectors in the $n$ -dimensional space $V$ therefore form a basis.
So $f_1, \dots, f_n$ is an orthonormal basis of eigenvectors of $T$ . This proves $P(n)$ , and by induction $P(n)$ holds for all $n$ .
The eigenvalues of these eigenvectors are real, by the theorem on eigenvalues.

Idea

Find one unit eigenvector $e$ , then look at the hyperplane orthogonal to it. Self-adjointness makes this hyperplane invariant, and $T$ stays self-adjoint there, so induction gives the rest of the basis. Over $\R$ , the eigenvalue comes from viewing the symmetric matrix over $\C$ : its complex eigenvalues are real, so a real eigenvector exists.

Reading path

To understand this page from the ground up, read these 629 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Steinitz exchange lemma by induction, Proof of the associativity of matrix multiplication, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that a subspace of full dimension is the whole space, Proof of the decomposition of a permutation into transpositions, Proof that the symmetric group is generated by transpositions, Proof of the Laplace expansion along a column by transposition, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the fundamental theorem of algebra by Liouville's theorem, Proof that equivalence classes form a partition, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the 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 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 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 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 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 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.

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Proof that equivalence classes form a partition proof
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Closed set definition
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Open cover definition
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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
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Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
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
Isomorphism with the coordinate space statement
Every spanning set contains a basis statement
Every linearly independent set extends to a basis statement
Approximation property of the supremum statement
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Decomposition of a permutation into transpositions statement
Proof of continuity via closed sets proof
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Closed sets contain their limits statement
Triangle inequality statement
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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 existence of an orthonormal basis by the Gram–Schmidt process proof
Proof of the intermediate value theorem by the supremum proof
Proof of Grassmann's formula by extending a basis of the intersection proof
Proof that the reals are Archimedean from the completeness axiom proof
Sign of a permutation definition
Continuity via closed sets statement
Modulus of a complex number definition
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof of the reverse triangle inequality proof
Proof that the symmetric group is generated by transpositions proof
Proof of the 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 of epsilon-delta continuity at a point in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof of continuity via open sets in metric spaces proof
Proof that metric spaces are Hausdorff proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of the linearity of the integral proof
Rank–nullity theorem statement
Proof that convergent sequences are bounded proof
Existence of an orthonormal basis statement
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Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Grassmann's formula statement
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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 that a continuous image of a compact set is compact proof
Proof of the sign of a cycle proof
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Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
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Proof that an injective endomorphism of a finite-dimensional space is surjective proof
Proof of the sequence characterisation of uniform continuity proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Linearity of the integral statement
Uniqueness of limits statement
Proof of the triangle inequality for norms proof
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Proof of the monotonicity of the integral proof
Subsequences of a convergent sequence converge to the same limit statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Proof that a space is the direct sum of a finite-dimensional subspace and its orthogonal complement proof
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
Continuous image of a compact set is compact statement
Proof of the Cauchy–Riemann equations proof
Sign is a homomorphism statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
An injective endomorphism of a finite-dimensional space is surjective statement
Epsilon-delta uniform continuity statement
Proof that permuting rows multiplies the determinant by the sign proof
Monotonicity of the integral statement
There is a natural number n with 1/n below any positive epsilon statement
Darboux and Riemann integrals agree statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
A space is the direct sum of a subspace and its orthogonal complement statement
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
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of Riemann's criterion for integrability proof
Proof of the formula for the dimension of the orthogonal complement proof
Modulus is multiplicative statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Riemann's criterion for integrability statement
Sequential characterization of limits statement
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
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
Product of continuous functions is continuous statement
Additivity of the integral over intervals statement
Chain rule statement
Ratio test statement
Proof of the Heine–Borel theorem by sequential compactness proof
Operator norm definition
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the trace is a coefficient of the characteristic polynomial by the Leibniz formula 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
Heine–Borel theorem statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Laplace expansion along a row statement
Determinant is multiplicative statement
Proof of the first fundamental theorem of calculus proof
Determinant of a transpose statement
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
Continuous real function on a compact set attains its bounds statement
Determinant of the inverse 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
Inverse in terms of the adjugate statement
Proof of the mean value theorem from Rolle's theorem proof
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

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Tag	`A81KPS16D7`
URI	`m2://A81KPS16D7`
Kind	proof
Writing	W0 stub
MSC	47B15, 15A18
Proves	Spectral theorem
Uses	Inner product, Self-adjoint operator, Eigenvalue, Eigenvector, Orthonormal basis, Existence of an orthonormal basis, Coordinates in an orthonormal basis, Existence of an eigenvalue over an algebraically closed field, Fundamental theorem of algebra, Algebraically closed field, Eigenvalues of a self-adjoint operator are real, Orthogonal complement, Dimension of the orthogonal complement, An orthonormal family is linearly independent, Linearly independent set of the right size is a basis
Changed	4 November 2026

Proof of the spectral theorem for self-adjoint operators by induction on the dimension

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