Symmetric Matrix and Its Properties

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. Mathematically, a matrix $A$ is symmetric if:

$$A=A^T$$

where $A^T$ is the transpose of $A$, meaning that the rows and columns of $A$ are interchanged.

Examples of Symmetric Matrices

Example 1: A 2×2 Symmetric Matrix

Consider the matrix:

$$A=\left[\begin{array}{cc}4& 2\\ 2& 3\end{array}\right]$$

Its transpose is:

$$A^T=\left[\begin{array}{cc}4& 2\\ 2& 3\end{array}\right]$$

Since $A=A^T$, the matrix is symmetric.

Example 2: A 3×3 Symmetric Matrix

$$B=\left[\begin{array}{ccc}5& -1& 3\\ -1& 2& 4\\ 3& 4& 1\end{array}\right]$$

Its transpose:

$$B^T=\left[\begin{array}{ccc}5& -1& 3\\ -1& 2& 4\\ 3& 4& 1\end{array}\right]$$

Since $B=B^T$, it is symmetric.

Properties of Symmetric Matrices

Property 1: The Sum of Two Symmetric Matrices is Symmetric

If $A$ and $B$ are symmetric matrices of the same size, then their sum $A+B$ is also symmetric.

Example

Let:

$$A=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right],B=\left[\begin{array}{cc}4& 5\\ 5& 6\end{array}\right]$$

Then:

$$A+B=\left[\begin{array}{cc}1+4& 2+5\\ 2+5& 3+6\end{array}\right]=\left[\begin{array}{cc}5& 7\\ 7& 9\end{array}\right]$$

Since $A+B=(A+B{)}^T$, the result is symmetric.

Property 2: The Scalar Multiple of a Symmetric Matrix is Symmetric

If $A$ is a symmetric matrix and $c$ is a scalar, then $cA$ is also symmetric.

Example

Let:

$$A=\left[\begin{array}{cc}2& 3\\ 3& 5\end{array}\right],c=3$$

Then:

$$3A=\left[\begin{array}{cc}6& 9\\ 9& 15\end{array}\right]$$

Since $(3A{)}^T=3A$, it remains symmetric.

Property 3: The Product of Two Symmetric Matrices is Symmetric if They Commute

If $A$ and $B$ are symmetric matrices and $AB=BA$, then $AB$ is also symmetric.

Example

Let:

$$A=\left[\begin{array}{cc}2& -1\\ -1& 3\end{array}\right],B=\left[\begin{array}{cc}4& 2\\ 2& 5\end{array}\right]$$

Calculate:

$$AB=\left[\begin{array}{cc}2(4)+(-1)(2)& 2(2)+(-1)(5)\\ -1(4)+3(2)& -1(2)+3(5)\end{array}\right]=\left[\begin{array}{cc}8–2& 4–5\\ -4+6& -2+15\end{array}\right]$$

$$=\left[\begin{array}{cc}6& -1\\ -1& 13\end{array}\right]$$

Since $AB=A{B}^T$, the product is symmetric.

Property 4: All Eigenvalues of a Symmetric Matrix are Real

If $A$ is a symmetric matrix, then all its eigenvalues are real numbers.

Example

Consider the symmetric matrix:

$$A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$$

Finding its eigenvalues involves solving:

$$det(A–\lambda I)=0$$

$$\left|\begin{array}{cc}2–\lambda & -1\\ -1& 2–\lambda \end{array}\right|=0$$

$$(2–\lambda )(2–\lambda )–(-1)(-1)=0$$

$${\lambda }^2–4\lambda +3=0$$

Solving for $\lambda$:

$$\lambda =\frac{4\pm \sqrt{16–12}}{2}=\frac{4\pm 2}{2}$$

$$\lambda =3,1$$

Both eigenvalues are real.

Property 5: A Symmetric Matrix is Diagonalizable

Every symmetric matrix is diagonalizable, meaning it can be expressed as:

$$A=PD{P}^{-1}$$

where $P$ is an orthogonal matrix and $D$ is a diagonal matrix containing eigenvalues of $A$.

Example

For:

$$A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$$

The eigenvalues are $\lambda =3,1$ (as found earlier). The corresponding eigenvectors form the columns of $P$, and the diagonal matrix $D$ consists of the eigenvalues:

$$D=\left[\begin{array}{cc}3& 0\\ 0& 1\end{array}\right]$$