Symmetric Matrices

Definition: A square matrix $A$ is said to Symmetric if $A = A^T$, that is for all entries in $A$, $a_{ij} = a_{ji}$.

We can imagine symmetric matrices to have entries that are mirror images of each other if we draw a line down the main diagonal:

For example, the following two matrices $A$ and $B$ are symmetric:

(1)

\begin{align} A = \begin{bmatrix} 1 & -2 & 0\\ -2 & 4 & 3\\ 0 & 3 & 2 \end{bmatrix} \quad , \quad B = \begin{bmatrix} 3 & 1\\ 1 & 4 \end{bmatrix} \end{align}

We will now look at a theorem outlining some important properties of symmetric matrices.

Theorem 1: If $A$ and $B$ are symmetric matrices of size $m \times n$ and $k \in \mathbb{R}$ is a scalar, then: a) $A^T$ is also symmetric. b) The sum $A + B$ and difference $A - B$ are also symmetric matrices. c) The matrix $kA$ is also symmetric. d) $A$ and $B$ being symmetric $\not \Rightarrow AB$ symmetry (i.e, $A$ and $B$ being symmetric does not imply that the product $AB$ is symmetric).

We will show all proofs except for (d). To show (d), all that is needed is a counter-example, that is two symmetric matrices that when multiplied doesn't produce a symmetric matrix.

• Proof of (a): Since $A^T$ results from interchanging the rows of $A$ with the columns of $A$, $A^T$ must also be symmetric. $\blacksquare$

• Proof of (b): Assume $A + B$ is symmetric, that is $A + B = (A + B)^T$. But we know that $(A + B)^T = A^T + B^T$. Since both $A$ and $B$ are already symmetric since $A = A^T$ and $B = B^T$, therefore $A + B = (A + B)^T$ and our initial assumption was true. $\blacksquare$

• Proof of (c): If $A$ is symmetric, multiplying each symmetric element in $A$ by $k$ will still have $kA$ being symmetric since every entry in the $A$ is multiplied by the same scalar. $\blacksquare$