Proving Eigenvalues Equality of A & B Matrices

[fluidistic]

Homework Statement

Let A=LU and B=UL, where U is an upper triangular matrix and L is a lower triangular matrix. Demonstrate that A and B have the same eigenvalues.

Homework Equations

Not sure.

The Attempt at a Solution

I know that if I can show that A and B are similar (so if I can find a matrix P such that P^(-1)AP=B) they have the same eigenvalues. But I didn't find P yet, nor do I know really how to search efficiently for P.

Another route I've thought of is to write det (I*lambda-A)=0 gives the same values for lambda as det (I*lambda-B)=0. I've thought of using det (A)=det (LU)=det L * det U = det U * det L = det B... but still can't reach anything I find useful.
Any tip is greatly appreciated.

 

[lanedance]

how about multiplying thorugh by the inverse of L.. you may need to show it exists

 

[zimo123]

Try to write A in terms of B (or the other way around), and see what you get!

 

[lanedance]

hint was probably enough

 

[fluidistic]

LU=U^(-1)ULU=LU so the equality is true. Notice that LU=A and UL=B. I have that P=U, thus LU is similar to UL (A similar to B) so they share the same eigenvalues.

Now I must justify the use of the inverse of U. Well I believe its element on the diagonal are all non zero. So if the matrix is nxn, the span of its column vector is R^n so it is invertible, hence U^-1 exists.


Is that well justified?

 

[Dick]

LU=U^(-1)ULU=LU so the equality is true. Notice that LU=A and UL=B. I have that P=U, thus LU is similar to UL (A similar to B) so they share the same eigenvalues.

Now I must justify the use of the inverse of U. Well I believe its element on the diagonal are all non zero. So if the matrix is nxn, the span of its column vector is R^n so it is invertible, hence U^-1 exists. Is that well justified?

It would be if the problem stated that the elements of U along the diagonal are nonzero. Does it? Other versions of this problem I've found say that L or U is a UNIT triangular matrix. Did you miss a word in the problem statement?

 

[fluidistic]

It would be if the problem stated that the elements of U along the diagonal are nonzero. Does it? Other versions of this problem I've found say that L or U is a UNIT triangular matrix. Did you miss a word in the problem statement?

You pointed out a very interesting thing to me. In my assignment it isn't specified. But most of the exercises assigned are taken from Kincaid's book on numerical analysis. I just checked there and it clearly states L to be UNITARY.
If only they had stated this in my assignment I could have guessed that I had to take the inverse of L. :) Well I'm not 100% sure but probably.
So basically the problem is solved now... thanks guys.

 

[Dick]

You pointed out a very interesting thing to me. In my assignment it isn't specified. But most of the exercises assigned are taken from Kincaid's book on numerical analysis. I just checked there and it clearly states L to be UNITARY.
If only they had stated this in my assignment I could have guessed that I had to take the inverse of L. :) Well I'm not 100% sure but probably.
So basically the problem is solved now... thanks guys.

Their bad. Very welcome.