Eigenvalues of Symmetric Matrix are Real

Corollary to Hermitian Matrix has Real Eigenvalues

Every real symmetric matrix has eigenvalues which are all real numbers.

Proof

We have that a Real Symmetric Matrix is Hermitian.

The result follows from Hermitian Matrix has Real Eigenvalues.

$\blacksquare$

Examples

Arbitrary Example

Let $\mathbf A$ be the real symmetric matrix defined as:

$\mathbf A = \begin {pmatrix} 1 & -2 & 5 \\ -1 & 6 & -1 \\ 5 & -2 & 1 \end {pmatrix}$

The eigenvalues of $\mathbf A$ are:

\(\ds \lambda_1\)

\(=\)

\(\ds -4\)

\(\ds \lambda_2\)

\(=\)

\(\ds 4\)

\(\ds \lambda_3\)

\(=\)

\(\ds 8\)

Their corresponding eigenvectors are:

\(\ds \mathbf v_1\)

\(=\)

\(\ds \begin {pmatrix} 2 \\ 0 \\ -2 \end {pmatrix}\)

\(\ds \mathbf v_2\)

\(=\)

\(\ds \begin {pmatrix} 1 \\ 1 \\ 1 \end {pmatrix}\)

\(\ds \mathbf v_3\)

\(=\)

\(\ds \begin {pmatrix} 1 \\ -1 \\ 1 \end {pmatrix}\)

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): eigenvalue