SME
Numerical Methods for Engineers
Eigenvalue methods
Use this for “Find the eigenvalues of A = [4 1; 2 3].”
Chart
Zeros: λ = 2, λ = 5
This is an example. Type your own values to start a new problem.
λ²
−
tr A
λ
+
det A
=
0
λ₁
=
λ₁
,
λ₂
=
λ₂
Δ
Discriminant
worked out (not typed)
Formulas
tr
A
=
a₁₁
+
a₂₂
7 = 4 + 3
det
A
=
a₁₁
×
a₂₂
−
a₁₂
×
a₂₁
10 = 4 × 3 − 1 × 2
Δ
=
(tr A)
2
−
4
×
(det
A)
9
=
(7)
2
−
4
×
10
λ₁
=
tr A +
√
Δ
2
5
=
7 +
√
9
2
λ₂
=
tr A −
√
Δ
2
2
=
7 −
√
9
2
λ₁
+
λ₂
=
tr
A
5 + 2 = 7
λ₁
×
λ₂
=
det
A
5 × 2 = 10
Assumptions
•
The characteristic equation det(A − λI) = λ² − (tr A)λ + det A = 0 gives the eigenvalues.
•
Real roots only: Δ = (tr A)² − 4 det A is at least 0.
•
λ₁ is the larger eigenvalue and λ₂ the smaller; they are equal when Δ = 0.
Step-by-step
WE KNOW
a₁₁ = 4, a₁₂ = 1, a₂₁ = 2, a₂₂ = 3
FIND
trace (tr A), determinant (det A), discriminant (Δ), larger eigenvalue (λ₁), smaller eigenvalue (λ₂)
Step 1 · Find trace (tr A)
Use tr A = a₁₁ + a₂₂
The trace is the sum of the diagonal entries.
tr A = 4 + 3
tr A = 7
Step 2 · Find determinant (det A)
Use det A = a₁₁ × a₂₂ − a₁₂ × a₂₁
Down the main diagonal minus up the other diagonal.
det A = 4 × 3 − 1 × 2
det A = 12 − 2
det A = 10
Step 3 · Find discriminant (Δ)
Use Δ = (tr A)² − 4 × (det A)
The
discriminant
of
λ
2
−
(tr
A)λ
+
det
A
=
0.
Below
0,
the
two
roots
are
complex.
Δ
=
(7)
2
−
4
×
10
Δ
=
49
−
4
×
10
Δ
=
49
−
40
Δ
=
9
Step 4 · Find larger eigenvalue (λ₁)
Use λ₁ = (tr A + √(Δ)) ÷ 2
The quadratic formula with the plus sign gives the larger root.
λ₁
=
7 +
√
9
2
λ₁
=
7 + 3
2
λ₁
=
10
2
λ₁
=
5
Step 5 · Find smaller eigenvalue (λ₂)
Use λ₂ = (tr A − √(Δ)) ÷ 2
The quadratic formula with the minus sign gives the smaller root.
λ₂
=
7 −
√
9
2
λ₂
=
7 − 3
2
λ₂
=
4
2
λ₂
=
2
Check
Put the numbers back in. Both sides should match.
7 = 4 + 3 ✓
10 = 4 × 3 − 1 × 2 ✓
9
=
(7)
2
−
4
×
10
✓
5
=
7 +
√
9
2
✓
2
=
7 −
√
9
2
✓
5 + 2 = 7 ✓
5 × 2 = 10 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.