What is Symmetric Matrix? Eigenvalues, Properties & Examples

Symmetric matrix is a square matrix P=[xij] in which (i, j)th element is similar to the (j, i)th element i.e. xij = xji for all values of i and j. In other words, a square matrix (P) which is equal to its transpose is known as symmetric matrix i.e. PT = P.

Symmetric matrix examples

The example of a symmetric matrix of order 2 (or matrix size is 2 x 2) is as follows.

    \[C=\begin{bmatrix} 1 &2 \\ 2& 6 \end{bmatrix}\]

The example of a symmetric matrix of order 3 (or matrix size is 3 x 3) is as follows.

    \[D=\begin{bmatrix} 1 &8 &2 \\ 8& 7 &5 \\ 2&5 & 9 \end{bmatrix}\]

Symmetric matrix properties

Let P and Q be symmetric matrices then

1. PT = P

2. QT = Q

3. P + Q is a symmetric matrix.

4. P – Q is a symmetric matrix.

5. PQ may or may not be a symmetric matrix.

6. QP may or may not be a symmetric matrix.

Let B be any matrix then

1. BBT is always a symmetric matrix.

2. \frac{B+B^T}{2}

is always a symmetric matrix.

Symmetric matrix eigenvalues

A symmetric matrix P of size n × n has exactly n eigen values. These eigen values is not necessarily be distinct. It is noted that there exist n linearly independent eigenvectors even if eigen values are not distinct. One eigen vector for each eigen value. These eigen vectors are mututally orthogonal.

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