Thread: find an othonormal set of eigenvectors

Find an orthonormal set of eigenvectors for the following symmetric matrix.

[ 10 0 2 ]
[ 0 6 0 ]
[ 2 0 7 ]

that's a 3x3 matrix, just don't know how to put it all in one.

Ok, where should you start? What have you done?

well i tried to find the characteristic equation,

and i got (x= lamda)
x^3 - 23x^2 - 168x - 396.. but then couldn't factorise it

Double-check your signs. I agree with the magnitudes of the coefficients, but I think you have a sign error in there somewhere.

ok, now i have , (x= lamda)

x^3 - 23x^2 + 168x - 396..

and i have (x-11)( x^2 -12x+36),

i think the second term is made up of two complex factors, but i forget how to do it?

Looking better. The quadratic factor you have there factors over the reals. In fact, it factors fairly straight-forwardly over the reals. You could always use the quadratic formula. What do you get?

lmao wow i can't believe i actually just did that. ( i swear i'm not this stupid, haha)
well, i have the eigeevalues now, i'l see how i go from here, thanks

Ok, let me know how it goes.

Originally Posted by linalg123

Find an orthonormal set of eigenvectors for the following symmetric matrix.

[ 10 0 2 ]
[ 0 6 0 ]
[ 2 0 7 ]

If you expand the determinant on the middle row you get which obviously has as a root.
Since the expansion you give obviously has 11 as a root, try factoring that quadratic as so that 6 is a double eigenvalue and 11 is an eigenvalue. Now start looking for eigenvectors.

(Double click on the formulas to see the LaTex code.)

that's a 3x3 matrix, just don't know how to put it all in one.

ok, i have two linearly independant eigenvectors. Is that a problem that i can't get a third?

Show us your work. It isn't necessarily a problem. It depends on whether you're required to get a basis of orthonormal eigenvectors. If you're just asked to get a set of orthonormal eigenvectors, then who cares how many there are in the set?

for \lamda =11

\left|\begin{array}{ccc}1 & 0 & -2 \\ 0 & 5 & 0\\ 2 & 0 & 4\end{array}\right|

y=0
x=2z eigenvector \left|\begin{array}{ccc}2 & 0 & 1\end{array}\right|

for \lamda=6

\left|\begin{array}{ccc}-4 & 0 & -2 \\ 0 & 0 & 0\\ -2 & 0 & -1\end{array}\right|

y=0
x=-2z eigenvector \left|\begin{array}{ccc}1 & 0 & -2\end{array}\right|

that look ok? so then i just make them orthonormal and i'm done?

that didn't work =S

It looks like you've correctly found two eigenvectors. (You should clean up the way that post looks by enclosing your LaTeX code with math tags.) However, you're missing an eigenvector.

for


y=0
x=2z eigenvector

for



y=0
x=-2z eigenvector

that look ok? so then i just make them orthonormal and i'm done?