Learn Linear Algebra

Theorem

If $A$ is symmetric, then any two eigenvectors from different eigenspaces are orthogonal.

Theorem

An $n \times n$ matrix $A$ is orthogonally diagonalizable if and only if $A$ is a symmetric matrix.

The Spectral Theorem for Symmetric Matrices

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

a. $A$ has $n$ real eigenvalues, counting multiplicities.
b. The dimension of the eigenspace for each eigenvalue $\lambda$ equals the multiplicity of $\lambda$ as a root of the characteristic equation.
c. The eigenspaces are mutually orthogonal, in the sense that eigenvectors corresponding to different eigenvalues are orthogonal.
d. $A$ is orthogonally diagonalizable.