Linear Algebra and Eigenvalues

[Bluesman01]

Suppose A is a diagonlizable nxn matrix where 1 and -1 are the only eigenvalues (algebraic multiplicity is not given). Compute A^2.

The only thing I could think to do with this question is set A=PD(P^-1) (definition of a diagonalizable matrix) and then A^2=(PD(P^-1))(PD(P^-1))=P(D^2)(P^-1)

This is how I left it on the test but I am sure this isn't right. How can you solve this without having the original matrix A or the algebraic multiplicity of the eigenvalues?

 

[Mark44]

Suppose A is a diagonlizable nxn matrix where 1 and -1 are the only eigenvalues (algebraic multiplicity is not given). Compute A^2.

The only thing I could think to do with this question is set A=PD(P^-1) (definition of a diagonalizable matrix) and then A^2=(PD(P^-1))(PD(P^-1))=P(D^2)(P^-1)

Since A is diagonalizable, and its only eigenvalues are 1 and -1, then what form must D take? Even more to the point, what does D² have to be?

This is how I left it on the test but I am sure this isn't right. How can you solve this without having the original matrix A or the algebraic multiplicity of the eigenvalues?

 

[Bluesman01]

But if the multiplicity of either eigenvalue is more than one, D changes form. Both P and D do. That's where I got stuck.

 

[Dick]

But if the multiplicity of either eigenvalue is more than one, D changes form. Both P and D do. That's where I got stuck.

Of course D changes form. D^2 doesn't. What is it? Once you figure out what it is, P won't matter.

 

[Bluesman01]

Ah I see it now. Thanks to all who replied.

 

[Mark44]

That would be all two of us: Dick and myself...