How to Prove that a Matrix is Positive Definite

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In this video I will teach you what a positive definite matrix is and how you can prove that a matrix is positive definite using the five fundamental properties of positive definite matrices. I will also show you a worked example of using Sylvester's criteria for positive definite matrices to actually prove that a matrix is in fact positive definite. I will also share with you a really simple way to instantly tell by looking at a matrix if it is likely to be positive definite or if it cannot be positive definite.

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Great explanation! Thank you! However, to the best of my knowledge, the Sylvester's criterion is a necessary and sufficient condition only for the symmetric matrices (or Hermitian matrices, if we include complex numbers). The final example was symmetric, but not the ones before that.

markmisin

I thought the +ve eigenvalues rule only works if your matrix is symmetric

kaiochanx

thank you very much, it was very useful, Allah bless you.

zinebadaika

@1:23 That is a weird definition of "symmetric". Usually, it's defined as A_(i, j) = A_(j, i), which seems like a way more natural way to define it. From that, I guess one can prove that there is some other matrix B such that A = B^T*B, but I don't think I've ever seen that before.

Furthermore, I think 4 can be stated much more simply as "the matrix is symmetric (and full rank)"

allyourcode

this is so useful, thanks for sharing the video!

whogashaga

I wish he would of explained a couple things.

1) What if there was a different number other than zero?
2) How was the last matrix created that have -1, -1, 0, & 2?

investwithvincent

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