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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.

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  • $\begingroup$ 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. $\endgroup$
    – Community Bot
    Feb 21 at 3:13
  • $\begingroup$ Note that $\|x\| = \|Ax\|$ and so the largest and smallest singular values are one. $\endgroup$
    – copper.hat
    Feb 21 at 4:08

1 Answer 1

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The reason is that "singular values" and "eigenvalues" are not the same thing.

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)

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Yahoooo
    Mar 2 at 13:51

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