I understand that the "singular values" used to form the diagonal matrix D in a singular value decomposition of a matrix A are the square roots of the eigenvalues of [itex]A^TA[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

But I'm having trouble getting a grip on the concept of "singularity" in the context of a matrix. Previously I've only come across singularity as a point where a function is non-continuous and therefore not differentiable. I don't see how that definition applies here. I read in at least one book that "a square matrix A is nonsingular if and only if all its singular values are different from zero." So it sounds like that means that 0 must NOT be an eigenvalue of [itex]A^TA[/itex]. (On second thought, this author is probably not saying that at all; rather just using "A" to represent several different matrices in the course of his discussion.)

But in another book, I got the distinct impression that an SVD for a matrix A can be formed just by using the nonzero singular values of A; i.e. if 0 is an eigenvalue of [itex]A^TA[/itex], just don't use 0; use the nonzero eigenvalues to form a smaller matrix D.

I also read that the condition number [itex]\frac{\sigma_1}{\sigma_n}[/itex] is a measure of thedegree of singularityof A.

But I have no feel at all for what is meant by "degree of singularity" of A, and what it meansqualitativelyfor a matrix to be singular or nonsingular.

Can anybody give aqualitativeexplanation of these terms?

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# Degree of singularity

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