Eigenvalues
The directions a matrix only stretches, and by how much.
Pen and paper is fine · no calculator needed why?
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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
- Trace: add the eigenvalues.
- Determinant: multiply them. Mind the signs.
worked 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
- 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
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
- 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
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
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.
practice
Trace & determinant
worked example
A 2×2 matrix has eigenvalues −2 and −4. What is its trace?
Answer: -6
- The trace is the sum of the eigenvalues: −2 − 4 = −6.
2 × 2 eigenvalues
worked example
What are the eigenvalues of [[2, −4], [−3, −2]]? Enter both, smaller first.
Answer: (-4, 4)
- Characteristic equation: λ² − 16 = 0 (trace 0, determinant −16).
- (λ + 4)(λ − 4) = 0, so the eigenvalues are −4 and 4.
Eigenvector
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
- Solve (A − 1I)v = 0 with v = (1, k). Its first row is (7 − 1, 6), so 6 × 1 + 6 × k = 0.
- That is 6 + 6k = 0, so k = −1.
- Check: A(1, −1) = (1, −1) = 1 × (1, −1).