What Is a Schur Decomposition?

Re:What Is a Schur Decomposition? by patrick451 on Monday March 04, 2024 @08:14PM (#39598251)

> For matrices, you can show these end up as finite series (i.e., it can be written with the highest n being the dimension of the matrix)

I don't think is quite accurate. It sounds like you may be referring to the Cayley-Hamilton theorem which says that A^n can be expressed as a linear combination of A^0, A^1,... A^n-1. But this is not the same as saying that the exponential taylor series is finite. Any 1x1 matrix is a counterexample. Another counter example is a diagonal matrix. It's exponential is the exponential of each diagonal element, but each of those requires an infinite series.

The statement is true for a certain class of matrices called nilpotent matrices. A nilpotent matrix has the property that for some k, A^k = 0. An example is

  A = [ 0, 1 ]
      [0   0 ]

In this case A * A =0, so the e^A clearly is a finite series.

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Re:What Is a Schur Decomposition? by cat_man on Tuesday March 05, 2024 @12:42AM (#39599849)

You're right, I did mistakenly mix a few concepts together. The series representation for the function is infinite, like you say. What I was misapplying was the idea that a convergent infinite sum of matrix products (c0I + c1A + ... + cnA^n + ...) can be written as finite polynomial, which relies on Cayley-Hamilton like you say. So f(A) for any particular choice of A, can be written as a finite polynomial, but the coefficients of that polynomial will change as A does.

So writing

exp(A) = I + A + A^2/2 + ... + A^n/n!

like I did isn't correct. The concept I was thinking of was a Taylor series like this

exp(A) = I + A + A^2/2 + ... + A^n/n! + ... (to infinity)

has an equivalent finite polynomial like:

exp(A) = c0I + c1A + ... +c_{n-1}*A^(n-1)

(usually expressed as order n-1, not n like I was said earlier). The c0, c1, ... c_{n-1} will depend on A, so not as generally useful as what I was misremembering. It does let you build representations for things like exp(tA) (where t is a scalar multiple) in terms of a finite polynomial (the terms depending on t, like the "exp(lambda*t)" in your diagonal matrix example, end up in the c0, c1, ..., c_{n-1} coefficients).

Thanks for the correction and sorry if my garbled attempt at explaining confused anyone. The point about matrix functions being based on series representations is independent of those mistakes.

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