A symmetric matrix is a matrix A such that

$$A^{t}=A $$

where \(A^{t}\) is a transposed matrix of A. Such a matrix is necessarily square. Its main diagonal entries are arbitrary, but its other entries occur in pairs on opposite sides of the main diagonal.

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Suppose that A is a \(3\times 3\) symmetric matrix:

$$A=\begin{bmatrix}a_{11} &a_{12}&a_{13}\\a_{21} &a_{22}&a_{23}\\a_{31} &a_{32}&a_{33}\\\end{bmatrix}$$

Since \(A^{t}=A\), we have

$$\begin{bmatrix}a_{11} &a_{12}&a_{13}\\a_{21} &a_{22}&a_{23}\\a_{31} &a_{32}&a_{33}\\\end{bmatrix}=\begin{bmatrix}a_{11} &a_{21}&a_{31}\\a_{12} &a_{22}&a_{32}\\a_{13} &a_{32}&a_{33}\\\end{bmatrix}$$

Therefore, a symmetric matrix is given as below:

$$A=\begin{bmatrix}a_{11} &a_{12}&a_{13}\\a_{12} &a_{22}&a_{23}\\a_{13} &a_{23}&a_{33}\\\end{bmatrix}$$

In general, a symmetric matrix is given as below:

$$A^{t}=A\ \ \Longleftrightarrow\ \ A=\begin{bmatrix}a_{11}&a_{12}&a_{13}&\cdots&a_{1n}\\a_{12} &a_{22}&a_{23}&\cdots&a_{2n}\\a_{13} &a_{23}&a_{33}&\cdots&a_{3n}\\\vdots &\vdots &\vdots& \ddots & \vdots\\a_{1n} &a_{2n}&a_{3n}&\cdots&a_{nn}\\\end{bmatrix}$$

That is

\begin{eqnarray*}&&a_{ii}\ \text{ : arbitary number } \ (\ i= j\ ) \\&&a_{ij}=a_{ji} \ (\ i\not= j\ )\end{eqnarray*}

where \(a_{ij}\) is \((i, j)\)-entry in a matrix A.

Example

\begin{eqnarray*}\text{Symmetric } \ &:& \ \begin{bmatrix}1&2\\2 &3\end{bmatrix}, \ \ \begin{bmatrix}1 &2&3\\2 &4&5\\3 &5&6\\\end{bmatrix}, \ \ \ \begin{bmatrix}1 &2&3&4\\2&5&6&7\\3& 6&8&9\\4&7&9&10\\\end{bmatrix}\\\text{Nonsymmetric } \ &:& \ \begin{bmatrix}1&2\\3 &4\end{bmatrix}, \ \ \begin{bmatrix}1 &2&3\\4&5&6\\7 &8&9\\\end{bmatrix}, \ \ \ \begin{bmatrix}1 &2&3&4\\5&6&7&8\\9& 10&11&12\\13&14&15&16\\\end{bmatrix}\end{eqnarray*}