# What is the singular value of such a simple permutation matrix?

The matrix $$\left[\begin{array} &0 &1\\1& 0\end{array}\right]$$

The following two solutions give different answers:

The first |A-λI|=0

(-λ) 1
1 (-λ)
= 0

∴(-λ)×(-λ)-1×1=0

∴(λ2)-1=0

∴(λ2-1)=0

∴(λ-1)(λ+1)=0

∴(λ-1)=0or(λ+1)=0

∴ The eigenvalues of the matrix A are given by λ=-1,1,

However, when I try to find the result in https://www.omnicalculator.com/math/singular-values it tell me the singular value is 1,1.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Feb 21 at 3:13
• Note that $\|x\| = \|Ax\|$ and so the largest and smallest singular values are one. Feb 21 at 4:08

You know what the eigenvalues of a matrix are. The singular values are the square roots of the eigenvalues of $$A^\ast A$$.
If $$A$$ is diagonal with non-negative eigenvalues, then these are the same thing, but in general, they are not the same (eg, singular values are always non-negative and real, but eigenvalues can be negative or complex)