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Eigenvalues of a 3x3 Matrix: Constants

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Find the eigenvalues of 3x3 square matrix of constants

3x3 square matrix of constants

Equation 1 .

How do we find the eigenvalues?

We find the characteristic equation of the matrix A.

Must the matrix be square?

Yes.

How do we find the characteristic equation?

First we find the determinant

Equation 2

This results in a polynomial involving

. The roots of this polynomial are the eigenvalues.

First, set up

for this problem.

Do the multiplication.

Add the matrices

Set up the determinant.

Equation 3

Set up the multiplication.

We get

Simplify in the square brackets.

We get

Factor the quadratic in the square brackets (this won’t always be possible, of course).

We get

characteristic equation of a matrix

Equation 4

Solve.

Any factor may be zero, so we get eigenvalues of a matrix

eigenvalues of a matrix

.

So what are the eigenvalues?

These values of

are the eigenvalues.

Checks Check Cayley-Hamilton Theorem appled to the matrix

Cayley-Hamilton Theorem appled to the matrix

in Equation 3.

Check

in Equation 3.

Check

in Equation 3.

Now let’s show that the Cayley-Hamilton Theorem is confirmed.
State this theorem.

One way to state it is that a square matrix satisfies its own characteristic equation.

In this problem, which (numbered) equation is the characteristic equation?

Equation 4.

Set up the matrix substitution.
Should the right-hand side be a scalar or a matrix?

It should be the 3x3 zero matrix: Cayley-Hamilton Theorem appled to the matrix

Cayley-Hamilton Theorem appled to the matrix

Equation 5

Substitute the values in A and I and combine the matrices in each set of parentheses.

Simplify the diagonal elements.

Equation 6

Multiply the first two matrices.

Now multiply this result by the third matrix in Equation 6.

or

We see that the Cayley-Hamilton Theorem is satisfied, giving us confidence that our characteristic equation and eigenvalues are correct.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

General Contents

Detailed Contents