<s>[INST] Singular Values of negative eigenvalue? Consider a $4 \times 4$ matrix with eigenvalues $0,1,-2,4$. What are the singular values of this matrix?
I am aware that singular values are just square root of the eigenvalues, but what will be the answer for $-2$ eigenvalue? Is it undefined?
 [/INST] @Alexander Vigodner provides the punchline
$$
 \sigma \left( \mathbf{A} \right) = \sqrt{\lambda \left( \mathbf{A} \right)}
$$
only when $\mathbf{A} = \mathbf{A}^{*}$.
We can provide a bit more insight with a smaller matrix. What are the $2\times 2$ matrices with the eigenvalue spectrum $\left( 1, 0 \right)$?
The characteristic polynomial is 
$$
 p(\lambda) = \lambda^{2} - \lambda.
$$
Symmetric matrices
The symmetric matrices have the form
$$
\mathbf{A}
=
\left[
\begin{array}{cc}
 a & b \\
 b & c \\
\end{array}
\right]
$$
with the characteristic equation:
$$
 p(\lambda) = \lambda^{2} - \lambda \, \text{tr } \mathbf{A} + \det \mathbf{A}
$$
The matrices can be expressed in terms of one parameter, $b$:
$$
\mathbf{A}(b)
=
\left[
\begin{array}{cc}
 \frac{1}{2} \left(1 \pm \sqrt{1-4 b^2} \right) & b \\
 b & 1 - \frac{1}{2} \left(1 \pm \sqrt{1-4 b^2} \right) \\
\end{array}
\right]
$$
Example:
$$
\mathbf{A}\left( 1 \right) = 
\left[
\begin{array}{cc}
 \frac{1}{2} \left(i \sqrt{3}+1\right) & 1 \\
 1 & \frac{1}{2} \left(1-i \sqrt{3}\right) \\
\end{array}
\right]
$$
Asymmetric matrices
Relax the symmetry requirement
$$
\mathbf{A}
=
\left[
\begin{array}{cc}
 a & b \\
 c & d \\
\end{array}
\right]
$$
$$
\mathbf{A}(b,c)
=
\left[
\begin{array}{cc}
 \frac{1}{2} \left(1 \pm \sqrt{1-4 b c} \right) & b \\
 c & 1 - \frac{1}{2} \left(1 \pm \sqrt{1-4 b c} \right) \\
\end{array}
\right]
$$
Example:
$$
\mathbf{A}\left( 1, -1 \right) = 
\left[
\begin{array}{cc}
 \frac{1}{2} \left(1+\sqrt{5}\right) & 1 \\
 -1 & \frac{1}{2} \left(1-\sqrt{5}\right) \\
\end{array}
\right]
$$
</s>