Linear Algebra 22i: Symmetric Matrices and the LDU Decomposition

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Oh my God, finally an LDU decomposition video that makes sense! Thank you so much!

WowMyNameIsUnique

With a little modification, there actually is a kind of cool relationship between LDL/Cholesky and eigenvalues.
If you have a symmetric positive definite matrix A and you want to compute the eigenvectors, you can swap the D from LDL with A and L with your initial eigenvector matrix (it can kinda be whatever, even the matrix itself, but helps if it's orthogonal). Then perform Choelsky decomposition (or gram schmidt or whatever) on the result, then right divide the result by the cholesky decomposition. This separates the eigenvectors out. You might need perform an additional iteration or two (it converges really fast), but when it converges, you'll have the eigenvalues on the diagonal of cholesky decomposition. It also automatically sorts the eigenvalues. It's really a beautiful way to do eigendecomposition.

cbbuntz

I really like your video. The explanations are so clear and easy to understand.

frezman

Thank you for the concise and clear explanations in your videos!

semioticlabs

you may find useful to look back at Linear Algebra 13e before viewing this one

denisgiard

Thank you so much sir.. you saved me!!

aayushisingh

Thanks, simple effective in such short time

tahoon

Hi professor, just one thing hope you can clarify, if LDL' is a must from original symmetric matrix? Thank you!

kevinshao

which video has the proof that in a symmetric matrix the L = U^T

lienz

LDU looks awesome but perhaps less useful.

debendragurung

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