A matrix is singular if and only if any of the eigenvalues are zero.

University of Michigan - Ann Arbor

Suppose that a matrix A has n linearly independent eigenvectors {$v^{(1)}$, . . . , $v^{(n)}$} with corresponding eigenvalues {$\lambda _1$, . . . , $\lambda _n$}. We may concatenate all the eigenvectors to form a matrix V with one eigenvector per column: V = [$v^{(1)}$ , . . . , $v^{(n)}$]. Similarly, we can get a vector containing eigenvalues, naming it as $\lambda$. The eigendecomposition of A is then given by
$A = Vdiag(\lambda)V^{-1}$

Linear Algebra - eigendecomposition

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Singular Matrix Judgement According to Eigenvalues

A matrix whose eigenvalues are all positive is called positive definite.

Positive Definite

A matrix whose eigenvalues are all positive or zero valued is called positive semidefi-
nite.

Positive Semidefinite

If all eigenvalues are negative, the matrix is negative definite.

Negative Definite

If all eigenvalues are negative or zero valued, it is negative semidefinite.

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