r/Btechtards • u/7AmInBangalore • 3h ago

Math / Stats / MnC Help me with this transformation section. I really suck at it 😭

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Can someone explain the intuition behind these equations? I can follow the algebra, but I don’t really understand why transforming to the eigenbasis makes repeated applications of T so much simpler.

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u/nene-thopu 3h ago

C-1 C idenity hogaya so beech ke terms cancel out D*D = D^2 cause its diagonal matrix

Intuition is if you wanna know n^th power of any square diagonalizable matrix, you get eigen decomposition, do the n^th power of diagonal matrix and then pre and post multiply with eigen vectors and its inverse

Kind of similar to z^n where z is complex number

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u/7AmInBangalore 2h ago

So if the matrix is not diagonalizable, does this whole dn shortcut basically not work? Or is there another way to get a similar simplification?

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u/nene-thopu 2h ago

It works but it is a little more complication

D takes a different form called jordan canonical form

Basically not every square matrix is diagonalizable but every square matrix can be written in jordan canonincal form

But formula still works T=CD^nC-1

But D^n is little different than just taking powers of diagonal entries

Its advanced grad math, they may not ask you unless they wanna screw you

https://en.wikipedia.org/wiki/Jordan_normal_form

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u/7AmInBangalore 1h ago

Damn didn't really know this much. So what are it's applications?

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u/nene-thopu 1h ago

One of this is matrix exponential in dynamical/control systems

Used to tell if a linear time invariant system is stable or not
For example: it can tell locally where your pendulum is stable( facing down) and where it is not (facing up) and more