Symmetric Matrices

Symmetric matrix. A square matrix

is called a symmetric matrix if $A^T = A$

. The main diagonal entries $a_{ii}$ 's are arbitrary, but its other entries $a_{ij}$ 's occur in pairs with the entries $a_{ji}$ 's of the opposite side of the main diagonal. That is,

is symmetric if $a_{ij}=a_{ji}$ holds for every pair

Theorem for eigenvectors. If is symmetric, then two eigenvectors corresponding to different eigenvalues are orthogonal.

Proof. Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be eigenvectors corresponding to the different eigenvalues $\lambda_1$ and $\lambda_2$ . By applying the symmetry of we obtain

$\displaystyle \lambda_1\langle\mathbf{v}_1,\mathbf{v}_2\rangle = (\lambda_1 \m... ...thbf{v}_2 = (A \mathbf{v}_1)^T \mathbf{v}_2 = \mathbf{v}_1^T A \mathbf{v}_2$

while we have

$\displaystyle \lambda_2\langle\mathbf{v}_1,\mathbf{v}_2\rangle = \mathbf{v}_1^T (\lambda_2\mathbf{v}_2) = \mathbf{v}_1^T A \mathbf{v}_2$

Since $\lambda_1 \neq \lambda_2$ , we conclude that $\langle\mathbf{v}_1,\mathbf{v}_2\rangle=0$ .

Orthogonal diagonalizability. Assume that the symmetric matrix is diagonalizable, that is, that the algebraic multiplicity is equal to the geometric multiplicity for each eigenvalue. Then we can find a orthonormal set $\{\mathbf{u}_1,\ldots,\mathbf{u}_n\}$ of eigenvectors, and construct the orthogonal matrix

$\displaystyle U = [\mathbf{u}_1\cdots\mathbf{u}_n]$

Since $U^{-1} = U^T$ , we obtain the diagonalization

$\displaystyle A = U D U^T$

with diagonal matrix

whose diagonal entries are the eigenvalues $\lambda_i$ 's corresponding to the eigenvectors $\mathbf{u}_i$ 's. Then

is said to be orthogonally diagonalizable.

Characterization of symmetric matrices. In fact, every symmetric matrix is orthogonally diagonalizable, which equivalently characterizes a symmetric matrix. That is, we have the following theorme.

Theorem. A square matrix is orthogonally diagonalizable if and only if is symmetric.

Spectral decomposition. Equivalently the symmetric matrix has a spectral decomposition

$\displaystyle A = \lambda_1\mathbf{u}_1\mathbf{u}_1^T + \cdots + \lambda_n\mathbf{u}_n\mathbf{u}_n^T$

with eigenvalues $\lambda_i$ and the corresponding eivenvectors $\mathbf{u}_i$ .

EXAMPLE 3. Let $A = \begin{bmatrix} 3 & -2 & 4 \\ -2 & 6 & 2 \\ 4 & 2 & 3 \end{bmatrix}$ be a symmetric matrix. Find orthonormal basis for each eigenspace corresponding the eigenvalue of $-2$ and $7$ . Orthogonally diagonalize the matrix .