Invertible matrix

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May 7th 2010, 11:54 AM
gralla55

Invertible matrix

I'm supposed to find the matrix P here. I've found the characteristic polynomial p(lamda) = -x - x^2 + x^3 + x^4 (just writing x instead of lamda). And I found the roots:

x=0
x=1
x=-1

But I can't make sense of what the book does next to find P... Any help would be very appreciated! =)
May 7th 2010, 12:19 PM
eigenvex

They don't tell you any more information? When you say "find the matrix P", do you mean find the 4x4 matrix with all of its values? Because I think that is impossible just given the characteristic polynomial. Multiple 4x4 matrices can yield that same polynomial. Also there are two -1 roots, just fyi...
May 7th 2010, 12:35 PM
Roam

Quote:

Originally Posted by gralla55

I'm supposed to find the matrix P here. I've found the characteristic polynomial p(lamda) = -x - x^2 + x^3 + x^4 (just writing x instead of lamda). And I found the roots:

x=0
x=1
x=-1

But I can't make sense of what the book does next to find P... Any help would be very appreciated! =)

Are you trying to diagonalize a given matrix? If so you must find n linearly independent eigenvectors of that nxn matrix, say $p_1,p_2,...,p_n$ and form the matrix $P=[p_1, p_2, ... p_n]$ . The matrix $P^{-1}AP$ will be diagonal and will have the eigenvalues corresponding to $p_1,p_2,...,p_n$ , respectively, as its successive diagonal entries.
May 7th 2010, 03:52 PM
dwsmith

This is a fourth degree polynomial. There needs to be four solutions. What was the matrix that gave you those lambda?

$\begin{bmatrix} a_{11}-\lambda & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22}-\lambda & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33}-\lambda & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44}-\lambda \end{bmatrix}$

To obtain your eigenvectors, you need to plug in each lambda and then rref the matrix. Then do that for the other eigenvalues until you have 4 eigenvectors. If you don't obtain 4 eigenvectors, this matrix isn't diagonalizable.
May 8th 2010, 04:48 AM
HallsofIvy

Please write the entire problem! I suspect the respondents are correct, that you are given a matrix, A, and are asked to find the matrix P such that $A= PDP^{-1}$ where D is a diagonal matrix having the eignvalues of A on its diagonal. But what P is depends strongly on what A is- and different matrices can have the same characteristic polynomial.

dwsmith- he did give all roots. -1 is a double root.
May 8th 2010, 08:47 AM
dwsmith

Quote:

Originally Posted by HallsofIvy

Please write the entire problem! I suspect the respondents are correct, that you are given a matrix, A, and are asked to find the matrix P such that $A= PDP^{-1}$ where D is a diagonal matrix having the eignvalues of A on its diagonal. But what P is depends strongly on what A is- and different matrices can have the same characteristic polynomial.

dwsmith- he did give all roots. -1 is a double root.

I understand but it should have been noted.