Linear Algebra - Find a matrix P that diagonalizes A

**saxophonemaster** · November 30th 2010, 08:10 PM

This question is broken up into 4 parts to help you get to the right answer. Below is where I am so far.

Find eigenvalues of A:
A=[2 13
-1 -12]

det(lI - A) = 0
So the equation is l^2-10l - 11=0
So lambda = -11,1

b) Find corresponding eigenvectors
if l=-11
eigenvector is [-1
1]
if l=1
Eigenvector is [-13
1]

c) Find a matrix P that diagonalizes A
P = [-1 -13
1 1]

d) Verify that P^-1 A P is diagonal
This verification does not work for me, so I assume I did something wrong above? Any idea where I went wrong?

**dwsmith** · November 30th 2010, 08:19 PM

With a 2x2, you don't have to use the det method.

The characteristic polynomial is simply $\displaystyle \lambda^2-tr(A)\lambda+det(A)\rightarrow \lambda^2+10\lambda-11=(\lambda+11)(\lambda-1)$

Your eigenvalues are correct. Looks correct to me though.

**dwsmith** · November 30th 2010, 08:33 PM

All your work is correct. You may have made a clerical error when doing the multiplication.

**FernandoRevilla** · November 30th 2010, 09:33 PM

Originally Posted by saxophonemaster

d) Verify that P^-1 A P is diagonal
This verification does not work for me, so I assume I did something wrong above? Any idea where I went wrong?

All is right. Verify:

$AP=\begin{pmatrix}{2}&{13}\\{-1}&{-12}\end{pmatrix}\begin{pmatrix}{-1}&{-13}\\{1}&{1}\end{pmatrix}=\begin{pmatrix}{11}&{-13}\\{-11}&{1}\end{pmatrix}$

$PD=\begin{pmatrix}{-1}&{-13}\\{1}&{1}\end{pmatrix}\begin{pmatrix}{-11}&{0}\\{0}&{1}\end{pmatrix}=\begin{pmatrix}{11}& {-13}\\{-11}&{1}\end{pmatrix}$

So, $P^{-1}AP=D$

Regards.

Fernando Revilla

**HallsofIvy** · November 30th 2010, 10:06 PM

Your eigenvalues and P are correct. What did you get for $P^{-1}$ ?

Thread: Linear Algebra - Find a matrix P that diagonalizes A

LinkBack

Thread Tools

Display

Linear Algebra - Find a matrix P that diagonalizes A

Similar Math Help Forum Discussions

Linear Algebra Matrix Question

Matrix that simultaneously diagonalizes A and B

linear algebra, rotation matrix

Linear Algebra Matrix Check

matrix representation of linear algebra

Search Tags