# Eigenspaces, Orthogonal + Diagonal Matrices

• May 14th 2010, 10:24 AM
kabirSarin
Eigenspaces, Orthogonal + Diagonal Matrices
Could someone please let me know how I would go about accomplishing the following tasks. Thank you.

Let A = [[4,0,0],[0,1,3],[0,3,1]

a) Find the Eigenvalues of A.

(lambda - 4)(lambda - 4)(lambda + 2)
So the eigenvalues are 4, 4, and -2.

b) For each eigenvalue, find a basis for the corresponding eigenspace.

??

c) Use these bases vectors to construct an orthogonal matrix P.

??

d) Verify that $\displaystyle P^-1AP$ is a diagonal matrix.

??
• May 14th 2010, 12:32 PM
dwsmith
Quote:

Originally Posted by kabirSarin
Could someone please let me know how I would go about accomplishing the following tasks. Thank you.

Let A = [[4,0,0],[0,1,3],[0,3,1]

a) Find the Eigenvalues of A.

(lambda - 4)(lambda - 4)(lambda + 2)
So the eigenvalues are 4, 4, and -2.

b) For each eigenvalue, find a basis for the corresponding eigenspace.

??

c) Use these bases vectors to construct an orthogonal matrix P.

??

d) Verify that $\displaystyle P^-1AP$ is a diagonal matrix.

??

Eigenvalues of A are

$\displaystyle \begin{bmatrix} 4-\lambda & 0 & 0\\ 0 & 1-\lambda & 3\\ 0 & 3 & 1-\lambda \end{bmatrix}\rightarrow (4-\lambda)[(1-\lambda)^2-9]=-(\lambda-4)^2(\lambda+2)$

When $\displaystyle \lambda=4$

$\displaystyle \begin{bmatrix} 0 & 0 & 0\\ 0 & -3 & 3\\ 0 & 3 & -3 \end{bmatrix}\rightarrow rref= \begin{bmatrix} 0 & 0 & 0\\ 0 & 1 & -1\\ 0 & 0 & 0 \end{bmatrix}$

$\displaystyle x_1$
$\displaystyle x_2=x_3$
$\displaystyle x_3$

$\displaystyle \begin{bmatrix} x_1\\ x_3\\ x_3 \end{bmatrix}\rightarrow x_1\begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}+x_3\begin{bmatrix} 0\\ 1\\ 1 \end{bmatrix}$

Now find the other eigenvector for lambda = -2

Once you have the third eigenvector, you can construct your P matrix that corresponds to A
• May 16th 2010, 03:42 AM
HallsofIvy
I prefer to use the definition of "eigenvector" directly: If 4 is an eigenvector for A, then there exist a vector, v, such that Av= 4v. Specifically,
$\displaystyle \begin{bmatrix}4 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 3 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \begin{bmatrix}4x \\ y+ 3z\\ 3y+ z\end{bmatrix}= 4\begin{bmatrix} x \\ y \\ z\end{bmatrix}$.

That gives the four equations 4x= 4x, y+ 3z= 4y, and 3y+ z= 4z. The first equation is satisfied by any x, of course and the last two are the same as 3y= 3z or z= y. That is, any eigenvector, corresponding to eigenvalue 4, is of the form $\displaystyle \begin{bmatrix}x \\ y \\ y\end{bmatrix}= x\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}+ y\begin{bmatrix}0 \\ 1 \\ 1\end{bmatrix}$, showing the two basis vectors for the eigenspace clearly.

With eigenvalue -2, those equations are simply 4x= -2x, y+ 3z= -2y, and 3y+ z= -2z. The first equation says x= 0 and I will leave the others to you.