General Contents
Detailed Contents
Index
Eigenvalues of a 2x2 Matrix: Constants
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Find the eigenvalues of
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.
We get br>
Equation 4
Factor (this won’t always be possible, of course).
We get
Solve.
Either factor may be zero, so we get
.
So what are the eigenvalues?
These values of
are the eigenvalues.
Checks:
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 vector?
It should be vector:
Equation 5
Next set up the determination of
Show terms and factors in the product.
Simplify.
We get
Substitute this result into Equation 5.
Do the multiplication.
Simplify.
We see that the Cayley-Hamilton Theorem is satisfied, giving us confidence that our characteristic equation and eigenvalues are correct.
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General Contents
Detailed Contents
Index
Equation 1
.