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.
@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)"
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?
@@TheCompleteGuide1 Ellipsoids (positive definite) are dual to hyperboloids (negative definite). Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature and hence gravitation is dual -- forces are dual. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Gaussian negative curvature is defined by at least two dual points -- non null homotopic. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. The Big Bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. "Always two there are" -- Yoda.
I thought the +ve eigenvalues rule only works if your matrix is symmetric
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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.
Exactly, he should have mentioned this while solving the example with Sylvester's method.
@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)"
thank you very much, it was very useful,Allah bless you.
Explanation is clear as daylight.
this is so useful, thanks for sharing the video!
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?
you are life saver man
Is it work for complex matrix?
thank you!
Very nice!
Awesomely explained!
Glad you think so! Often advanced maths is very badly explained.
@@TheCompleteGuide1 Ellipsoids (positive definite) are dual to hyperboloids (negative definite).
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature and hence gravitation is dual -- forces are dual.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Gaussian negative curvature is defined by at least two dual points -- non null homotopic.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
The Big Bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero -- non null homotopic.
"Always two there are" -- Yoda.