What do the singular values of a matrix tell you?

That may not be very clear, but for example let’s say I have a matrix whose elements are a function of 2 variables x, y. When I do SVD of this matrix over 10 iterations of values for (x,y), the SVD of each iteration will be different. The way I understand it is, the values on the diagonal of the sigma matrix, if 0 (or close to it) then we know that the matrix we are decomposing is singular. For the matrix to be singular (or nearly singular), do all the values on that diagonal have to be very small, or only one?

Furthermore, if we want to know the values of x, y that caused the singularity then we simply check what they were for that iteration. Is this correct logic?

Sorry if this is a poorly formed question, math is really not my strong suit, but I’m trying to reason through this problem nonetheless. Math is hard.

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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