91Ó°ÊÓ

• A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
• For example, is a two-row, three-column matrix

If A is symmetric matrix, then ${\mathrm{A}}^2$is also a symmetric matrix. As matrix A is symmetric, from the spectral theorem there exist an orthogonal matrix and a diagonal matrix D such that:

$\mathrm{A}={\mathrm{SDS}}^{-1}$

As the matrix S is invertible, we have role="math" localid="1659607343924" $\mathrm{rank}\left(\mathrm{A}\right)=\mathrm{rank}\left(\mathrm{D}\right)$

Now evaluate ${\mathrm{A}}^2$as represented below:

$$\begin{gathered} {\mathrm{A}}^2=\mathrm{AA} \\ =\left({\mathrm{SDS}}^{-1}\right)\left({\mathrm{SDS}}^{-1}\right) \\ =\mathrm{SD}\left({\mathrm{S}}^{-1}\mathrm{S}\right){\mathrm{DS}}^{-1}⇒{\mathrm{A}}^2={\mathrm{SD}}^2{\mathrm{S}}^{-1} \end{gathered}$$

As the matrix S is invertible we get $\mathrm{rank}\left({\mathrm{A}}^2\right)=\mathrm{rank}\left({\mathrm{D}}^2\right)$.

If D is a diagonal matrix, as a result we get $\mathrm{rank}\left(\mathrm{D}\right)=\mathrm{rank}\left({\mathrm{D}}^2\right)$.

Hence we have as follows:

$$\begin{gathered} \mathrm{rank}\left(\mathrm{A}\right)=\mathrm{rank}\left(\mathrm{D}\right) \\ =\mathrm{rank}\left({\mathrm{D}}^2\right)\left(\mathrm{Since}\mathrm{rank}\left(\mathrm{D}\right)=\mathrm{rank}\left({\mathrm{D}}^2\right)\right) \\ ⇒\mathrm{rank}\left(\mathrm{A}\right)=\mathrm{rank}\left({\mathrm{A}}^2\right) \end{gathered}$$

Hence, we can conclude that, if A is a symmetric matrix, then $\mathrm{rank}\left(\mathrm{A}\right)=\mathrm{rank}\left({\mathrm{A}}^2\right)$.