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The singular values \(\{\sigma_1, \sigma_2, \dots, \sigma_m\}\) of a matrix \(A\) can be defined in several ways:

• the square roots of the eigenvalues of \(B=A^\star A\text{,}\)
• the lengths of the semiaxes of the hyperellipse that is the image of the unit circle under the linear transformation defined by \(A\text{.}\)
• The diagonal entries of the diagonal matrix \(\Sigma\) when we write \(A\) has a factorisation \(A=U \Sigma V^\star\) where $U$ and $V$ are unitary matrices, and $\Sigma$ is a diagonal matrix. (We'll be a little more precise below).