# What is the difference between “singular value” and “eigenvalue”?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is.

Is “singular value” just another name for eigenvalue?

The singular values of a $M\times N$ matrix $X$ are the square roots of the eigenvalues of the $N\times N$ matrix $X^*\,X$ (where $^*$ stands for the transpose-conjugate matrix if it has complex coefficients, or the transpose if it has real coefficients).
Thus, if $X$ is $N\times N$ real symmetric matrix with non-negative eigenvalues, then eigenvalues and singular values coincide, but it is not generally the case!