Properties of Symmetric Matrices

Definition¹

An arbitrary square matrix $A$ is called a symmetric matrix if it satisfies the following equation.

$$ A = A^{\mathsf{T}} $$

$A^{\mathsf{T}}$ is the transpose of $A$.

Properties

(a) $A^{\mathsf{T}}$ is a symmetric matrix.

(b) $A \pm B$ is a symmetric matrix.

(c) $kA$ is a symmetric matrix.

(d) If $A$ is invertible, then $A^{-1}$ is also a symmetric matrix.

(e) If $A$ is invertible, then $A^{\mathsf{T}}A$ and $AA^{\mathsf{T}}$ are also invertible.

(f) $AA^{\mathsf{T}}$ is a $m \times m$ symmetric matrix, and $A^{\mathsf{T}}A$ is a $n \times n$ symmetric matrix.

(g) All eigenvalues of $A$ are real.

(h) Eigenvectors of $A$ corresponding to distinct eigenvalues are orthogonal. (= Eigenvectors from different eigenspaces are orthogonal.)

Proof

(d)

Assume $A$ is an invertible matrix. Then $(A^{\mathsf{T}})^{-1} = (A^{-1})^{\mathsf{T}}$, hence $A^{-1}$ is also a symmetric matrix.

■

(e)

Let $A$ be an $m \times n$ matrix. Then the size of $AA^{\mathsf{T}}$ is $(m \times \cancel{n} ) \times (\cancel{n} \times m) = m \times m$ and, by the properties of the transpose matrix, the following holds.

$$ (AA^{\mathsf{T}})^{\mathsf{T}} = (A^{\mathsf{T}})^{\mathsf{T}} (A)^{\mathsf{T}} = AA^{\mathsf{T}} $$

Therefore $AA^{\mathsf{T}}$ is a symmetric matrix. The proof for $A^{\mathsf{T}}A$ is similar.

■

(f)

By the properties of invertible matrices, if $A$ is invertible then $A^{\mathsf{T}}$ is also invertible, and since the product of invertible matrices is invertible, $AA^{\mathsf{T}}$ and $A^{\mathsf{T}}A$ are invertible as well.

■

(g), (h)

These can be viewed as specializations of the proofs for Hermitian matrices.

• Eigenvalues of a Hermitian matrix are real
• Eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal

■