courses › Linear Algebra

Eigenvalues

level 52 course

The directions a matrix only stretches, and by how much.

Pen and paper is fine · no calculator needed why?

Opens at level 52.

Builds on: Inverses & rank (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

An eigenvector of a matrix A is a direction that A only stretches: Av = λv. The stretch factor λ (lambda) is its eigenvalue. Most vectors get turned; eigenvectors stay on their own line.

Two facts tie the eigenvalues to numbers you can read off the matrix. Their sum is the trace, the sum of the diagonal entries. Their product is the determinant. For a 2 × 2 matrix that is enough to find them: they solve λ² − (trace)λ + (determinant) = 0.

Techniques

Sum is trace, product is determinant

You know the eigenvalues and need the trace or the determinant.

Trace: add the eigenvalues.
Determinant: multiply them. Mind the signs.

worked example

Example: A 3 × 3 matrix has eigenvalues 4, −2 and 3. What is its determinant?

Multiply: 4 × (−2) × 3 = −24.
The trace would be 4 − 2 + 3 = 5.

Answer: −24

2 × 2 eigenvalues

Both eigenvalues of a 2 × 2 matrix [[a, b], [c, d]].

Trace T = a + d. Determinant D = ad − bc.
Solve λ² − Tλ + D = 0: find two numbers that add to T and multiply to D.
Enter the smaller one first.

worked example

Example: What is the larger eigenvalue of [[5, 2], [3, 4]] (rows listed)?

Trace 5 + 4 = 9. Determinant 5 × 4 − 2 × 3 = 14.
Two numbers that add to 9 and multiply to 14: 2 and 7.
The larger is 7.

Answer: 7

Eigenvector from the first row

An eigenvector of the form (1, k) for a known eigenvalue λ.

Subtract λ from both diagonal entries to get A − λI.
Its first row times (1, k) must be 0: (a − λ) × 1 + b × k = 0.
Solve for k. It may be a decimal, like 0.5.
Check: A times (1, k) should equal λ times (1, k).

worked example

Example: A = [[4, 1], [2, 3]] (rows listed) has eigenvalue 2. An eigenvector for 2 has the form (1, k). What is k?

A − 2I = [[2, 1], [2, 1]].
First row: 2 × 1 + 1 × k = 0, so k = −2.
Check: A times (1, −2) is (2, −4), which is 2 times (1, −2).

Answer: −2

Tips by skill

TipTrace & determinant: The trace is the sum of the eigenvalues; the determinant is their product.
Tip2 × 2 eigenvalues: Find two numbers that add to the trace and multiply to the determinant. Enter the smaller first.
TipEigenvector: Subtract λ from the diagonal. The first row times (1, k) must be 0: solve that for k.

Watch out for

Adding the eigenvalues when asked for the determinant. The sum is the trace; the product is the determinant.
Reading the diagonal entries as the eigenvalues. That is safe only when everything on one side of the diagonal is 0.
Forgetting to subtract λ from both diagonal entries before solving for k.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Trace & determinant par 16 s not tried yet
worked example
A 2×2 matrix has eigenvalues −2 and −4. What is its trace?

Answer: -6
1. The trace is the sum of the eigenvalues: −2 − 4 = −6.
2 × 2 eigenvalues par 60 s not tried yet
worked example
What are the eigenvalues of [[2, −4], [−3, −2]]? Enter both, smaller first.

Answer: (-4, 4)
1. Characteristic equation: λ² − 16 = 0 (trace 0, determinant −16).
2. (λ + 4)(λ − 4) = 0, so the eigenvalues are −4 and 4.
Eigenvector par 40 s not tried yet
worked example
A = [[7, 6], [−5, −4]] (rows listed) has eigenvalue 1. An eigenvector for 1 has the form (1, k). What is k? Enter a whole number or a decimal.

Answer: -1
1. Solve (A − 1I)v = 0 with v = (1, k). Its first row is (7 − 1, 6), so 6 × 1 + 6 × k = 0.
2. That is 6 + 6k = 0, so k = −1.
3. Check: A(1, −1) = (1, −1) = 1 × (1, −1).

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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