# Eigenvalues

• Nov 3rd 2007, 08:01 PM
Thomas
Eigenvalues
Find all the eigenvalues (real and complex) of the matrix:

$\displaystyle \left( \begin{array}{ccc} -6 & 22 & 11 \\ -2 & -7 & -2 \\ -1 & 26 & 10 \end{array} \right)$

I started off by using the formula for finding the characteristic polynomial, and reducing the determinant matrix to:

$\displaystyle det\left( \begin{array}{ccc} 0 & 26x+134 & -x^2+4x+49 \\ 0 & x+59 & -2x+22 \\ 1 & -26 & x-10 \end{array} \right)$

Which then got me to:

$\displaystyle det\left( \begin{array}{ccc} 26x+134 & -x^2+4x+49 \\ x+59 & -2x+22 \end{array} \right)$

When I take the determinant of that, I get a cubic polynomial which is quite difficult to solve.

Is there an easier way to do this, or am I just doing it completely wrong? Can I possibly reduce it more?
• Nov 3rd 2007, 08:13 PM
kalagota
Quote:

Originally Posted by Thomas
Find all the eigenvalues (real and complex) of the matrix:

$\displaystyle \left( \begin{array}{ccc} -6 & 22 & 11 \\ -2 & -7 & -2 \\ -1 & 26 & 10 \end{array} \right)$

I started off by using the formula for finding the characteristic polynomial, and reducing the determinant matrix to:

$\displaystyle det\left( \begin{array}{ccc} 0 & 26x+134 & -x^2+4x+49 \\ 0 & x+59 & -2x+22 \\ 1 & -26 & x-10 \end{array} \right)$

Which then got me to:

$\displaystyle det\left( \begin{array}{ccc} 26x+134 & -x^2+4x+49 \\ x+59 & -2x+22 \end{array} \right)$

When I take the determinant of that, I get a cubic polynomial which is quite difficult to solve.

Is there an easier way to do this, or am I just doing it completely wrong? Can I possibly reduce it more?

$\displaystyle P_x(A) = |xI - A|$
what is your characteristic polynomial again?
• Nov 3rd 2007, 08:20 PM
Thomas
I found the characterisitic polynomial to be:

$\displaystyle x^3+3x^2+19x+57$
• Nov 3rd 2007, 08:23 PM
Jhevon
Quote:

Originally Posted by Thomas
I found the characterisitic polynomial to be:

$\displaystyle x^3+3x^2+19x+57$

provided that is correct, i did not check it, we know that x = -3 is one root. thus, by the factor theorem (x + 3) is a factor. use long division to determine the others (the other two are complex)
• Nov 3rd 2007, 08:28 PM
Thomas
I'm quite confident it's correct - I re-did all the steps about three times and have arrived at the same polynomial. :p

To find one root is x=-3, did you just use a graphing calculator?
• Nov 3rd 2007, 08:29 PM
Jhevon
Quote:

Originally Posted by Thomas
I'm quite confident it's correct - I re-did all the steps about three times and have arrived at the same polynomial. :p

To find one root is x=-3, did you just use a graphing calculator?

no. you check the factors of the constant, that is, the factors of 57 (consider the negative and positive factors) and plug them in to see if you get zero, this method is by the rational roots theorem
• Nov 3rd 2007, 08:29 PM
kalagota
Quote:

Originally Posted by Thomas
I found the characterisitic polynomial to be:

$\displaystyle x^3+3x^2+19x+57$

did u notice that x=-3 is a factor?? Ü
• Nov 3rd 2007, 08:31 PM
Thomas
Ahh, I didn't remember that. Thank you. I'll post what I get for my values once I'm done!
• Nov 3rd 2007, 08:43 PM
Thomas
So I get $\displaystyle x^2+19=0$

Can anyone refresh my memory on complex numbers? :)

Would the answer be $\displaystyle x=+/-\sqrt{19}\:i$?
• Nov 3rd 2007, 08:45 PM
Jhevon
Quote:

Originally Posted by Thomas
So I get $\displaystyle x^2+19=0$

Can anyone refresh my memory on complex numbers? :)

Would the answer be $\displaystyle x=+/-\sqrt{19}\:i$?

yes, it is $\displaystyle \pm \sqrt{19}~i$
• Nov 3rd 2007, 08:46 PM
Thomas
Awesome, now wish me luck trying to enter this into my online assignment. :p
• Nov 11th 2007, 05:08 PM
DavidB
Quote:

Originally Posted by Thomas
Find all the eigenvalues (real and complex) of the matrix:

. . .

When I take the determinant of that, I get a cubic polynomial which is quite difficult to solve.

Is there an easier way to do this, or am I just doing it completely wrong? Can I possibly reduce it more?

Hi, Thomas.

Your answers are correct (-3, +/- 4.35889894354067i); however, I am wondering if you needed algebraic answers. (Just curious why you didn't use MATLAB or an online tool for calculating numeric results.)

David
• Nov 11th 2007, 07:28 PM
Thomas
I'm not sure why I didn't... I just did it the long way I guess. :p

As for algebraic answers... the online assignment program is set to allow answers within 1%, so it's fine handing in answers with decimals.
• Nov 12th 2007, 12:28 AM
Opalg
Quote:

Originally Posted by Thomas
Find all the eigenvalues (real and complex) of the matrix:

$\displaystyle \left( \begin{array}{ccc} -6 & 22 & 11 \\ -2 & -7 & -2 \\ -1 & 26 & 10 \end{array} \right)$

Just for future reference, these questions are often rigged so that you can take factors out of the determinant (by doing elementary row and column operations) without having to multiply it out. In this case, the 22 and the 11 in the top row of the matrix suggest that you might try subtracting twice the right column from the middle column. Then you get

$\displaystyle \begin{vmatrix}-6-x & 22 & 11 \\ -2 & -7-x & -2 \\ -1 & 26 & 10-x \end{vmatrix} = \begin{vmatrix}-6-x & 0 & 11 \\ -2 & -3-x & -2 \\ -1 & 6+2x & 10-x \end{vmatrix}$.

When you see that, you know that you have struck lucky, because there is a factor 3+x going down the middle column. Take this factor out, and you are left with $\displaystyle (3+x)\begin{vmatrix}-6-x & 0 & 11 \\ -2 & -1 & -2 \\ -1 & 2 & 10-x \end{vmatrix}$. Now add twice the middle row to the bottom row, and expand down the middle column, to get $\displaystyle (3+x)\begin{vmatrix}-6-x & 11 \\ -5 & 6-x \end{vmatrix} = -(3+x)(x^2 - 36 + 55) = -(3+x)(x^2 + 19)$. There's the eigenvalue equation, neatly factorised, without having to do any heavy calculations at all!