Linear Algebra 22i: Symmetric Matrices and the LDU Decomposition

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[@MathTheBeautiful]

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[@WowMyNameIsUnique]

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

[@cbbuntz]

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.

[@frezman2009]

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

[@MathTheBeautiful]

Thank you for letting me know.

[@semioticlabs9211]

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

[@xaviert7224]

Very nice explanation! Thank you!

[@denisgiard130]

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

[@aayushisingh7909]

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

[@tahoon2009]

Thanks , simple effective in such short time

[@kevinshao9148]

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

[@MathTheBeautiful]

Yes. A symmetric matrix always leads to LDL'

[@kevinshao9148]

@@MathTheBeautiful Thank you so much!

[@militantpacifist4087]

Thank you! That was really helpful!

[@zgwenwen]

thank you so much...... it is soooo beautiful!!!

[@lienz7302]

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

[@blazerboy466]

Thank you helped me out a lot!

[@debendragurung3033]

LDU looks awesome but perhaps less useful.

[@picosdrivethru]

does A = LU work as S = LU???????????????????

[@picosdrivethru]

such as S = LU where I used gauss-jordan to solve for L.. it does NOT work??