Orthogonally diagonalizing the matrix

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In summary, the conversation discusses the process of orthogonally diagonalizing a matrix by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D. The conversation also mentions finding eigenvectors and eigenspaces, and the benefits of diagonalizing a matrix.
  • #1
war485
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Homework Statement



This is for linear algebra/matrix:

Orthogonally diagonalize this matrix A by finding an orthogonal matrix Q and a diagonal matrix D such that QTAQ = D

A =
[ 1 2 2 ]
[ 2 1 2 ]
[ 2 2 1 ]

Homework Equations



(A - [tex]\lambda[/tex]I ) = 0

The Attempt at a Solution



D =
[5 0 0 ]
[0 -1 0 ]
[0 0 -1 ]

characteristic equation : -[tex]\lambda[/tex]3 + [tex]\lambda[/tex]2 + 9[tex]\lambda[/tex] + 5 = 0

[tex]\lambda[/tex] = 5, -1, -1 (I got these after factoring the characteristic equation)

when [tex]\lambda[/tex] = 5, I got v1 = [ 1 1 1 ]

Then I'm almost done but I got stuck when trying to find v2 and v3 when [tex]\lambda[/tex] = -1 because when I tried to do it, it turned out weird (it turned into a zero matrix!):
[ 0 0 0 ]
[ 0 0 0 ]
[ 0 0 0 ]

So I think it means that x1 , x2 and x3 are all free variables for v2 and v3 , but if that's the case, then how can I make v1 v2 v3 into an orthogonal matrix if they're not independent?? I almost got it but I've no idea what to do now! Does this mean that it is not possible to orthogonally diagonalize it?
 
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  • #2
Ok, I'm pretty sure I got it but I still have a problem.
[edit] sorry for making it complicated earlier.
I'll dumb down my problem:

I need help seeing that this matrix
[ 1 1 1 ]
[ 1 1 1 ]
[ 1 1 1 ]

have these two eigenvectors:
[-1, 1, 0]

[-1, 0, 1]

how?
I keep getting [ 0 -1 -1 ] and [ -1 0 -1 ]
 
  • #3
war485 said:
have these two eigenvectors:
[-1, 1, 0]

[-1, 0, 1]

how?
I keep getting [ 0 -1 -1 ] and [ -1 0 -1 ]
It is impossible for a matrix to have exactly two eigenvectors. Instead, it might have a two-dimensional space of eigenvectors...

(incidentally, it's very easy to check if a given vector is an eigenvector...)
 
  • #4
maybe I used the wrong terminology.
I think I meant that one of its eigenspace is the span of { [-1, 1, 0] , [-1, 0, 1] }
but I can't see how.

But I can see that its other eigenspace is [ 1 1 1 ]
 
  • #5
First, I claim that it's very easy to show that that span is a subspace of the -1 eigenspace, just by direct verification.


Secondly, I was trying to give you a hint by making you use more precise terminology. The problem is to find a particular vector space. Your answer key specified a basis for some vector space. Your work computed a basis for some vector space. You're focusing too much on the fact that your basis is different than the answer key's basis... but you haven't spent any effort checking whether or not the answer key's vector space is equal to or different from your vector space...

If you're given spanning sets for two vector spaces, how do you check if they're equal or not?
 
  • #6
Why orthogonally diagonalize a matrix?
 
  • #7
matqkks said:
Why orthogonally diagonalize a matrix?
Because your teacher requires it on homework or a test?

But there are many good reasons to diagonalize a matrix- diagonal matrices are far easier to work with that other matrices- it becomes easy to take any power, find the exponential, or, generally, any function that has a Taylor's series.

"Orhogonally" diagonalizing a matrix is not quite as important but any matrix that can be diagonalized can be diagonlized using orthogonal matrices. And orthogonal matrices are relatively easy to handle.
 

Related to Orthogonally diagonalizing the matrix

What does it mean to "orthogonally diagonalize a matrix"?

Orthogonal diagonalization is a process in linear algebra where a square matrix is transformed into a diagonal matrix using an orthogonal matrix. This means that the resulting diagonal matrix will have all non-zero values on the main diagonal and zeros everywhere else.

Why is orthogonal diagonalization important?

Orthogonal diagonalization is important because it simplifies the calculation of matrix powers, matrix exponentials, and matrix logarithms. It also allows for easier solving of systems of linear differential equations and finding eigenvalues and eigenvectors.

What is the difference between diagonalization and orthogonal diagonalization?

Diagonalization is the process of transforming a square matrix into a diagonal matrix, while orthogonal diagonalization involves transforming a matrix into a diagonal matrix using an orthogonal matrix. This means that in orthogonal diagonalization, the resulting diagonal matrix will have orthogonal rows and columns.

Can any matrix be orthogonally diagonalized?

Yes, any square matrix can be orthogonally diagonalized if it has a complete set of n linearly independent eigenvectors. However, not all matrices can be diagonalized using a general diagonalization process.

What are the applications of orthogonal diagonalization?

Orthogonal diagonalization has numerous applications in fields such as engineering, physics, and computer science. It is used in solving systems of differential equations, analyzing vibrations in mechanical systems, and in data compression algorithms.

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