Question

The eigenvalues of a symmetric matrix $\mathbf{A}$must be equal to the singular values of$\mathbf{A}$.

Short answer

The given statement is FALSE.

Step-by-step solution

Find the eigenvalues

Consider a matrix

$\mathrm{A}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

Determine the eigenvalues as follows:

$\left(\mathrm{A}-\mathrm{\lambda l}\right)=0$

$$\begin{gathered} \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]-\left[\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right]=\left[\begin{array}{cc}1-\lambda & 1\\ 0& -1-\lambda \end{array}\right] \\ \left(1-\lambda \right)\left(-1-\lambda \right)=0 \\ -1-\lambda +\lambda +{\lambda }^2=0 \\ {\lambda }^2-1=0 \\ \lambda =\pm 1 \end{gathered}$$

Thus, the eigenvalues are $1,-1$.

Find the singular values.

Find ${\mathrm{A}}^{\mathrm{T}}\mathrm{of}\mathrm{A}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$as follows:

${\mathrm{A}}^{\mathrm{T}}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

Find ${\mathrm{A}}^{\mathrm{T}}\mathrm{A}$ as follows:

$$\begin{gathered} {\mathrm{A}}^{\mathrm{T}}\mathrm{A}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right] \\ {\mathrm{A}}^{\mathrm{T}}\mathrm{A}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \end{gathered}$$

Determine the singular values as follows:

$\left({\mathrm{A}}^{\mathrm{T}}\mathrm{A}-\mathrm{\lambda I}\right)=0$

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

$$\begin{gathered} \left(1-\lambda \right)\left(1-\lambda \right)=0 \\ {\left(1-\lambda \right)}^2=0 \\ {\left(-\lambda +1\right)}^2=0 \\ \lambda =1 \end{gathered}$$

The singular values are found from the non-zero eigenvalues of ${\mathrm{A}}^{\mathrm{T}}\mathrm{A}$.

The singular value is $\mathrm{\sigma }=\sqrt{\mathrm{\lambda }}$.

Thus, the singular value of A is 1.

Final Answer

In view of the above example, the eigenvalues of a symmetric matrix A are not equal to A's singular value.

The given statement is FALSE.