📂Matrix Algebra

Symmetric Matrices, Skew-Symmetric Matrices

Definition¹

A square matrix $A$ is called a symmetric matrix if it satisfies the following equation:

$$A=A^{T}$$

Here, $A^{T}$ is the transpose of $A$. $A$ is called an anti-symmetric matrix if it satisfies the following equation:

$$A =-A^{T}$$

Explanation

By the definition of the transpose, matrices that are not square cannot be symmetric or anti-symmetric. If $A$ is an anti-symmetric matrix, it follows from the definition that $a_{ii}=-a_{ii}$, so the diagonal elements must be $0$.

Properties

Let $A$ and $B$ be symmetric matrices of the same size, and let $k$ be an arbitrary constant.

(a) $A^{T}$ 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) Let $A$ be the matrix $m \times n$. Then $AA^{T}$ is an $m \times m$ symmetric matrix, and $A^{T}A$ is an $n \times n$ symmetric matrix.

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

Proof

(d)

Let $A$ be an invertible matrix. Then $(A^{T})^{-1} = (A^{-1})^{T}$ applies, and thus $A^{-1}$ is also a symmetric matrix.

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(e)

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

$$(AA^{T})^{T}=AA^{T}$$

Therefore, $AA^{T}$ is a symmetric matrix. The proof for $A^{T}A$ is the same.

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(f)

By the properties of invertible matrices, if $A$ is invertible, then $A^{T}$ is also invertible, and the product of invertible matrices is invertible, therefore $AA^{T}$, $A^{T}A$ are also invertible.

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Theorem

The necessary and sufficient condition for the product of two matrices to be symmetric is that the product of the two matrices is commutable.

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Keep in mind that the product of two matrices is generally not commutable.