Math Is Fun Forum

Hixy

Member

Registered: 2011-09-24

Posts: 15

Eigenvalues and similarity

The problem is as follows:
Given two matrices, A and B (see below),
a) State for any a all eigenvalues for both A and B.
b) Give for both A and B the algebraic and geometric multiplicities for all of the eigenvalues.
c) For which values of a are A and B, respectively, similar to a diagonal matrix?
d) For which values of a are A and B mutually similar?

Note: By "similar" I mean as in similarity between matrices (http://en.wikipedia.org/wiki/Similar_matrix)

My answers:
a) By finding the characteristic polynomial, i.e.

, we see that the eigenvalues are 1, a and a. However, when

, we have the eigenvalue 1 occurring three times. When

, we have 1 occurring as an eigenvalue once, and a twice.

Is this correct? Am I missing something about the a 'complexity'?

b) Here it's simply just finding the eigenvectors and giving their number for the geometric multiplicity for

and

.

c) Matrix B can only be similar to the 3x3 diagonal matrix with the entries 1, -1 and 1. Thus a in A must be -1. Right?

d) What is the best way to check for matrices being similar? I know that similar matrices can be thought of as describing the same linear map with respect to different bases. Thus I've checked for rank, determinant, trace, eigenvalues, the characteristic polynomial and given the respective values for a for each of these to equal each other for A and B. Is this the way to go?

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Eigenvalues and similarity

eigenvalues can appear more than once, as far as I know.

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X'(y-Xβ)=0

khdew

Member

Registered: 2011-12-11

Posts: 1

Re: Eigenvalues and similarity

to answer your question d: the best way to find out whether two matrices A and B are similar or not is by finding the modal matrix P that relates both matrices: B = inverse(P)*A*P. If the modal matrix P exist, you can be sure that A and B are similar. Modal matrix can be obtained from the concatenation of eigenvectors of matrix A. I found out Kardi Teknomo's tutorial (please Google it) on Linear Algebra give a reasonable good explanation with worked out numerical examples.

You can solve the above problem easily using Computer Algebra System like Mathematica or Maple, or using the free one like Maxima (again you can Google it since the forum does not allow me to post link).

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Eigenvalues and similarity

Hi;

You can download maxima as part of the geogebra program at.

http://code.google.com/p/geogebra/downloads/list

There you will find the latest version.

Or you can use Sage which includes maxima.

http://www.sagemath.org/

Or you can take the problem over to Wolfram Alpha where some of mathematica is online for everyone.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.