- #1

- 30

- 0

## Main Question or Discussion Point

I'm required to diagonalize a 5x5 matrix by hand, using 'appropriate similarity transformations.' I'm not asking for an answer to a homework question, I'm asking is there any sort of easier (by hand) way of doing this than how I'm approaching it?

If it matters, the matrix is as follows:

A= {

{1, 2, 1, 0, 0},

{2, 1, 2, 0, 0},

{1, 1, 2, 0, 0},

{0, 0, 0, 0, 2},

{0, 0, 0, 2, 0}}

Looking at this makes it seem like a 3x3 matrix, with a 2x2 tacked on the bottom right corner, and zero's added to the filler space made as a result of increasing by 2 dimensions.

The method I'm planning on using to diagonalize this:

- Find eigenvalues, not sure how many there are, but I know there could be 5 max

- Find normalized eigenvectors associated with each eigenvalue

- Use the rule: A is diagonalize-able where P

--- P is a matrix where each column is an eigenvector of A

This seems WAY too complicated to me. The class this is for is supposed to be a class about the mathematical methods in Physics, and this just seems like Linear Algebra all over again. This has no physical meaning to me and is just manipulating matrices using certain properties of matrices to get other matrices.

So, any thoughts on how to approach this? I vaguely remember someone else commenting on how the 5x5 looks like a 2x2 tacked on a 3x3, and somehow using that to our advantage to do this. But I seriously don't want to find the eigenvectors of a 5x5 by hand the way I've been doing it for a 3x3.

If anyone can explain what is going on physically here as well, that would be great. I want to understand this stuff, but right now it's just way too mathy.

If it matters, the matrix is as follows:

A= {

{1, 2, 1, 0, 0},

{2, 1, 2, 0, 0},

{1, 1, 2, 0, 0},

{0, 0, 0, 0, 2},

{0, 0, 0, 2, 0}}

Looking at this makes it seem like a 3x3 matrix, with a 2x2 tacked on the bottom right corner, and zero's added to the filler space made as a result of increasing by 2 dimensions.

The method I'm planning on using to diagonalize this:

- Find eigenvalues, not sure how many there are, but I know there could be 5 max

- Find normalized eigenvectors associated with each eigenvalue

- Use the rule: A is diagonalize-able where P

^{-1}AP = D (D being diagonal matrix)--- P is a matrix where each column is an eigenvector of A

This seems WAY too complicated to me. The class this is for is supposed to be a class about the mathematical methods in Physics, and this just seems like Linear Algebra all over again. This has no physical meaning to me and is just manipulating matrices using certain properties of matrices to get other matrices.

So, any thoughts on how to approach this? I vaguely remember someone else commenting on how the 5x5 looks like a 2x2 tacked on a 3x3, and somehow using that to our advantage to do this. But I seriously don't want to find the eigenvectors of a 5x5 by hand the way I've been doing it for a 3x3.

If anyone can explain what is going on physically here as well, that would be great. I want to understand this stuff, but right now it's just way too mathy.