Finding Eigenvalues

The characteristic equation, and where it comes from.

12 problems · about 25 minutes · N-VM.C.11, A-REI.B.4

What this lesson teaches

The student computes eigenvalues from the characteristic equation and finds eigenvectors.

Av = λv rearranges to (A − λI)v = 0, which forces det(A − λI) = 0.
For a 2 by 2 the characteristic equation is λ² − (trace)λ + (determinant) = 0.
Eigenvalues sum to the trace and multiply to the determinant, which makes checking easy.

Straightforward practice. Get the method working first.

5 problems

Trace 12 and determinant 35. What is the larger eigenvalue?
Answer 7
Why 7.
Why must the determinant of A − λI be zero?
Answer It has to crush a non-zero vector, and only a singular matrix does that.
Why A non-trivial null space forces a zero determinant.
Rows [3, 0] and [0, 8]. What is the trace?
Answer 11
Why 3 + 8.
Rows [3, 0] and [0, 8]. What is the larger eigenvalue?
Answer 8
Why A diagonal matrix wears its eigenvalues openly.
Trace 10 and determinant 21. What is the larger eigenvalue?
Answer 7
Why 7 and 3.

The same ideas with more to keep track of.

3 problems

Eigenvalues 6 and 2. What is the determinant?
Answer 12
Why They multiply to it.
Eigenvalues 6 and 2. What is the trace?
Answer 8
Why They sum to it.
Trace 0 and determinant 1. Are the eigenvalues real? 1 yes, 0 no.
Answer 0
Why The discriminant 0 − 4 is negative.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps for finding eigenvalues and eigenvectors.
Answer 1. Write the characteristic equation from the trace and determinant 2. Solve it for the eigenvalues 3. Substitute each eigenvalue back into A minus lambda I 4. Solve that system for the eigenvector
Why The eigenvalues must be known before the vectors can be found.
A triangular matrix with diagonal 2 and 9. What is the smaller eigenvalue?
Answer 2
Why The diagonal entries are the eigenvalues.
The Trace: Rows [6, 2] and [1, 5]. What is the trace?
Answer 11
Why 11.
The Missing One: Trace 9 and one eigenvalue is 4. What is the other?
Answer 5
Why 5.