Spectral Theorem

by Qiang Gao, Mar 20, 2017

Theorem (the spectral theorem)

An $n \times n$ symmetric matrix $\mathbf{A}$ has the following properties:

1. $\mathbf{A}$ has $n$ real eigenvalues, counting the multiplicities of the root of the characteristic equation (algebraic multiplicities).

2. The dimension of the eigenspace (geometric multiplicities) for each eigenvalue $\lambda$ equals the algebraic multiplicities.

3. The eigenspaces are multually orthogonal.

4. $\mathbf{A}$ is orthogonally diagonalizable.

Proof

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