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I'm deficient in some understanding about eigenvectors/values and diagonalisation and singular values. I came to a result with the following matrix
$$ A = \begin{bmatrix} 2 & 4 \\ -4 & 2 \\ \end{bmatrix} $$

To find the singlur values I find the roots of the characteristic equation.
$det(\lambda I-A^TA)=0$
This comes to
$$ \begin{vmatrix} \lambda -20 & 0 \\ 0 & \lambda - 20 \\ \end{vmatrix} = 0 $$

Which actually gives $\lambda_{1} = 20$ and $\lambda_{2}=20$.

Now this is where my knowledge is lacking. I don't know why they are called "singular" values. So to me if $\delta_{1} = \sqrt {20}$ and $\delta_2 = \sqrt {20}$. My question is, can this be true? Does singular value mean that each eigenvalue has to be distinct and each singular value has to be distinct? Can we have two singular values the same? Or is "singular" referring to something else?

Thanks.

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The term "singular value" has nothing to do with how many of them there are. You are allowed to have repeated singular values. Apparently, "singular value" was simply the way that "eigenvalue" was translated in a particular context, and we've stuck with the terminology ever since.

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  • $\begingroup$ Thanks @Omnomnomnom $\endgroup$ – Bucephalus Sep 16 '17 at 3:18

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