Question

Find the dimension of the space of all symmetric nxn matrices.

Short answer

The dimension of a nxn symmetric matrices is $\frac{n^2+n}{2}$which is spanned by

$Span\left\{\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],...,\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right]\right\}$.

Step-by-step solution

Determine the basis.

Consider the matrix $A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{12}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \dots & 2_{nn}\end{array}\right]$where all $a_{ij}$are real.

The matrix A is symmetric if ${\mathit{A}}^{\mathbf{T}}\mathbf{=}\mathit{A}$and the general form of any skew-symmetric matrix is role="math" localid="1660131384364" $\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{12}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \dots & 2_{nn}\end{array}\right]$.

The total number of elements in the upper triangular matrix is $\frac{{\mathbf{n}}^{\mathbf{2}}\mathbf{+}\mathbf{n}}{\mathbf{2}}$.

Simplify the equation role="math" localid="1660131413444" $A=\left[\begin{array}{cccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& \dots & {\mathrm{a}}_{1\mathrm{n}}\\ {\mathrm{a}}_{12}& {\mathrm{a}}_{22}& \dots & {\mathrm{a}}_{2\mathrm{n}}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{a}}_{1\mathrm{n}}& {\mathrm{a}}_{2\mathrm{n}}& \dots & 2_{\mathrm{nn}}\end{array}\right]$as follows.

role="math" localid="1660132125768" $$\begin{gathered} A=\left[\begin{array}{cccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& \dots & {\mathrm{a}}_{1\mathrm{n}}\\ {\mathrm{a}}_{12}& {\mathrm{a}}_{22}& \dots & {\mathrm{a}}_{2\mathrm{n}}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{a}}_{1\mathrm{n}}& {\mathrm{a}}_{2\mathrm{n}}& \dots & 2_{\mathrm{nn}}\end{array}\right] \\ A=\left[\begin{array}{cccc}{a}_{11}& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right]+\left[\begin{array}{cccc}0& a_{12}& \dots & 0\\ a_{12}& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right]+...+\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& a_{(n-1)n}\\ 0& 0& a_{(n-1)n}& 0\end{array}\right]+\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& a_{nn}\\ 0& 0& 0& 0\end{array}\right] \\ A=a_{11}\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right]+a_{12}\left[\begin{array}{cccc}0& 1& \dots & 0\\ 1& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right]+...a_{\left(n-1\right)n}\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right]+a_{nn}\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 1\end{array}\right] \\ \mathrm{where}\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],\left[\begin{array}{cccc}0& 1& \dots & 0\\ 1& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],...,\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 1\end{array}\right]\mathrm{are}\mathrm{linear}\mathrm{independent}. \end{gathered}$$

.

Therefore, the matrix A is spanned by .

$Span\left\{\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],\left[\begin{array}{cccc}0& 1& \dots & 0\\ 1& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],...,\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 1\end{array}\right]\right\}$

By the definition, the total number of elements in the setis .

$\left\{\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],\left[\begin{array}{cccc}0& 1& \dots & 0\\ 1& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],...,\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 1\end{array}\right]\right\}\mathrm{is}\frac{{\mathrm{n}}^2+\mathrm{n}}{2}$

Hence, the dimension of A is $\frac{{\mathrm{n}}^2+\mathrm{n}}{2}$which is spanned by

$Span\left\{\left[\begin{array}{cccc}1& 0& \dots & 0\\ 0& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],\left[\begin{array}{cccc}0& 1& \dots & 0\\ 1& 0& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & 0\end{array}\right],...,\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\left[\begin{array}{cccc}0& \dots & 0& 0\\ ⋮& \ddots & 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 1\end{array}\right]\right\}$.