# Eigenspaces and eigenvectors

• Dec 1st 2009, 12:18 PM
noles2188
Eigenspaces and eigenvectors
Find all 2 x 2 matrices A for which E(sub 7) = R^2.

Does E(sub 7) mean there are 7 eigenspaces? Can someone clarify this problem further?
• Dec 1st 2009, 01:14 PM
tonio
Quote:

Originally Posted by noles2188
Find all 2 x 2 matrices A for which E(sub 7) = R^2.

Does E(sub 7) mean there are 7 eigenspaces? Can someone clarify this problem further?

I think it'd be a good idea YOU first find out what is $E_7$ since it is not, apparently, standard notation. Ask you book/teacher.

Tonio
• Dec 1st 2009, 06:14 PM
noles2188
According to my book:

E(sub 7) = ker(A - 7I(sub 2)) = {v[IMG]file:///C:/Users/Geoffrey/AppData/Local/Temp/moz-screenshot.png[/IMG]in R^2 : Av = 7v}

7 is an eigenvalue and, to me, the question is asking to find matrices A such that Av = 7v. How would I go about this?
• Dec 1st 2009, 11:12 PM
lepton
$E_7$ denotes the eigenspace associated with the eigenvalue 7.
• Dec 2nd 2009, 02:40 AM
tonio
Quote:

Originally Posted by noles2188
According to my book:

E(sub 7) = ker(A - 7I(sub 2)) = {v[IMG]file:///C:/Users/Geoffrey/AppData/Local/Temp/moz-screenshot.png[/IMG]in R^2 : Av = 7v}

7 is an eigenvalue and, to me, the question is asking to find matrices A such that Av = 7v. How would I go about this?

Ok then, let $A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right )$ be such a matrix, then $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)\ left(\begin{array}{c}x\\y\end{array}\right)=\left( \begin{array}{c}7x\\7y\end{array}\right)\,\,\foral l\,x\,,\,y\in \mathbb{R}$ , since they want $E_7=\mathbb{R}^2$.

Well, now choose wisely $x\,,\,y\in\mathbb{R}$ to find the coefficients of the matrix. For example, choosing $x=1\,,\,y=0$ gives us at once that $a=7\,,\,c=0$...

Tonio