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Proof of the Spectral Theorem for Real Symmetric Matrices

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 1.

Theorem

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 2.

Proof by Induction

We prove the Spectral Theorem for real symmetric matrices by induction on the dimension n.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 3.

Base Case

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 4.

Induction Hypothesis

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 5.

Inductive Step: The n × n Case

Find the Characteristic Polynomial of A

The characteristic polynomial of A is used to establish the existence of an eigenvalue and corresponding eigenvector.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 6.
Recall the Fundamental Theorem of Algebra
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 7.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 8.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 9.
Show that λ1 is Real for a Symmetric Matrix

Since A is real and symmetric, we show that the eigenvalue λ1 must be real.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 10.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 11.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 12.
Normalize the Eigenvector of λ1
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 13.
Construct the Orthogonal Complement

We construct Q2 with orthonormal columns spanning the subspace orthogonal to q1.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 14.
Show the Columns of Q2 are Invariant Under A

We show that Aqk ∈ Col(Q2) for every column qk of Q2, establishing that the column space of Q2 is invariant under A.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 15.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 16.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 17.
Restrict A to the Column Space of Q2

We define B = Q2TAQ2, representing the restriction of A to the (n − 1) dimensional subspace Col(Q2).

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 18.
Show that B is Symmetric
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 19.
Show that Q2B = AQ2
Handwritten Spectral Theorem proof slide recalling that Q₂Q₂ transpose is the projection matrix onto the column space of Q₂ and showing that Q₂Q₂ transpose A Q₂ equals A Q₂ because each A qₖ lies in the column space of Q₂.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 21.
Apply the Induction Hypothesis

Since B is a real symmetric (n − 1) × (n − 1) matrix, the induction hypothesis applies to B.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 22.
Find the Remaining Eigenvectors of A
Handwritten Spectral Theorem proof slide finding the remaining eigenvectors of A using the eigenvectors of B and the matrix Q₂.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 24.
Show W has Orthonormal Columns
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 25.
Define the Matrices of Eigenvectors and Eigenvalues of A
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 26.
Show Q is Orthogonal
Handwritten Spectral Theorem proof slide recalling that q1 is a unit vector, W has orthonormal columns, and q1 is orthogonal to every column of Q2.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 28.
Show A = QDQT

Combining the orthonormal eigenvectors of A gives the orthogonal diagonalization A = QDQT, completing the proof of the Spectral Theorem.

Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 29.
Handwritten proof slide for the Spectral Theorem for real symmetric matrices, slide 30.
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