Excelent example of the importamce of null spaces ... all too often assumed to be comprised of only the zero vector. In one of the early lectures on Tesors you had an exercise about the positive definiteness of a matrix. I conviniently skipped the exercise but now I will have to go back and do it.
This is true when working with real numbers, but in practice it's usually necessary to use computers and floating point numbers. In that case roundup errors can cause eigenvalues to go negative.
Thanks for your comment. What you are saying sounds reasonable, but I wonder if this is common.Obviously, roundoff error can make a positive number negative. But I imagine it would be a challenge, or at least a fun problem, to find a floating-precision vector x and positive definite matrix A, such that xᵀAx < 0. I'm not an expert in numerical linear algebra and I'd love to see an example of this. I think A would need to be [ 1 1 - eps 1-eps 1 ]
@@MathTheBeautiful I think this happens when you have a high dimensional matrix with a lot of zero eigenvalues. Roundup causes the values to go very slightly negative. www.value-at-risk.net/non-positive-definite-covariance-matrices/
If you have a matrix that is positive definite, not only can you be sure that all of its eigenvalues are positive, but also that there's exactly one Cholesky decomposition.
That's a fantastic question. Michael's answer points to the very nice features of positive definite matrices. I will provide application examples in future videos!
en.wikipedia.org/wiki/Lionel_Messi. If you go through his videos on Lem.ma, you'll find that nearly every time he says "messy" he'll sneak in a frame of the famous soccer player.