Tips on finding the eigenvalues of a 3x3 matrix

In summary, calculating the eigenvalues of a 3x3 matrix, while tedious, can be made easier by using row operations and looking for simple roots in the characteristic polynomial. The goal is to understand what an eigenvalue is, not necessarily to solve cubic equations. Teachers and books will often provide examples with simple roots for this purpose.
  • #1
kamil
6
0
I find it rather tedious to calculate the eigenvalues of a 3x3 matrix. For example
[tex]The \emph{characteristic polynomial} $\chi(\lambda)$ of the
3$3 \times 3$~matrix
\[ \left( \begin{array}{ccc}
1 & -1 & -1 \\
-1 & 1 & -1 \\
-1 & -1 & 1 \end{array} \right)\]
is given by the formula
\[ \chi(\lambda) = \left| \begin{array}{ccc}
1-\lambda & -1 & -1 \\
-1 & 1-\lambda & -1 \\
-1 & -1 & 1-\lambda \end{array} \right|.\] [/tex]

Now if I do this by develloping the minors I get a cubic equation and I can't solve it without at least 30 minutes. I find it time consuming, especially during an exam.
 
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  • #2
One tip is that row operations can help you solve for the characteristic polynomial. Remember, given the matrix [tex] A [/tex], however, perform row operations on the matrix [tex]A - \lambda I[/tex], and not just [tex] A [/tex] in order for it to be valid.
 
  • #3
The first thing to remember is that you got this question from a teacher or a book (possibly both) in all likelihood. They're not trying to make you really good at solving cubic equations (for which there is a formula by the way) or test your abilities to do something not particularly interesting like searching hard for solutions - they're testing that you understand what an eigenvalue is.

Thus one can deduce that there will almost always be simple roots of the equations in the examples they set you. Have you tried plugging in some integers near to zero to the characteristic polynomial? Just by looking at the matrix one can see that -1 is an eigenvalue (there is an obvious candidate for an eigenvector, the column vector of 1s, equivalently the sum of the entries in each row is -1).
 

What is an eigenvalue of a 3x3 matrix?

An eigenvalue of a 3x3 matrix is a number that is associated with a specific set of eigenvectors. It represents the scaling factor by which the eigenvectors are multiplied when the matrix is applied to them.

Why is it important to find the eigenvalues of a 3x3 matrix?

Finding the eigenvalues of a 3x3 matrix is important because it allows us to understand the behavior of the matrix when it is applied to different vectors. It also helps in solving systems of linear equations and in analyzing the stability of dynamic systems.

What is the process of finding the eigenvalues of a 3x3 matrix?

The process of finding the eigenvalues of a 3x3 matrix involves finding the characteristic polynomial of the matrix, which is a polynomial equation that relates the eigenvalues to the matrix elements. The eigenvalues can then be found by solving this polynomial equation.

Is there a shortcut or easier method for finding the eigenvalues of a 3x3 matrix?

Yes, there is a shortcut method called the determinant method. This involves calculating the determinant of the matrix and then solving a quadratic equation to find the eigenvalues. This method is faster and simpler than the traditional method of finding the characteristic polynomial.

Can the eigenvalues of a 3x3 matrix be complex numbers?

Yes, the eigenvalues of a 3x3 matrix can be complex numbers. This is because the characteristic polynomial can have complex roots, and the eigenvalues are the roots of this polynomial. In fact, even if the matrix has all real elements, it can still have complex eigenvalues.

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