SME
Linear Algebra
Eigenvalues and eigenvectors
Use this for “Find the eigenvalues and eigenvectors of [[4, 1], [2, 3]].”
Diagram
A = [[4, 1], [2, 3]] sends (1, 0) to (4, 2) and (0, 1) to (1, 3): its columns.
Area of the image = |det A| × area of the figure: |4 × 3 − 1 × 2| × 1 = 10.
The unit circle becomes an ellipse.
A(1, 1) = (5, 5) = 5(1, 1): the line along (1, 1) maps onto itself, stretched by λ = 5.
A(1, −2) = (2, −4) = 2(1, −2): the line along (1, −2) maps onto itself, stretched by λ = 2.
This is an example. Type your own values to start a new problem.
A
=
λ₁
=
λ₁
,
v₁
=
1
λ₂
=
λ₂
,
v₂
=
1
tr A
Trace of A
worked out (not typed)
det A
Determinant of A
worked out (not typed)
Δ
Discriminant tr² − 4 det
worked out (not typed)
Formulas
tr
A
=
a
+
d
7 = 4 + 3
det
A
=
a
×
d
−
b
×
c
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
p₁
=
λ₁
−
a
b
1
=
5 − 4
1
p₂
=
λ₂
−
a
b
−2
=
2 − 4
1
Assumptions
•
Av = λv means A only stretches v: the line along v maps onto itself, scaled by λ.
•
det(A − λI) = 0 is λ² − (tr A)λ + det A = 0. Real eigenvalues need its discriminant to be 0 or more.
•
Each eigenvector is written v = (1, p), and any multiple of it is one too. This needs b ≠ 0.
Step-by-step
WE KNOW
a = 4, b = 1, c = 2, d = 3
FIND
trace of A (tr A), determinant of A (det A), discriminant tr² − 4 det (Δ), larger eigenvalue (λ₁), smaller eigenvalue (λ₂), slope of v₁ (p₁), slope of v₂ (p₂)
Step 1 · Find trace of A (tr A)
Use tr A = a + d
The trace is the sum of the diagonal entries.
tr A = 4 + 3
tr A = 7
Step 2 · Find determinant of A (det A)
Use det A = a × d − b × c
Down the main diagonal, minus the other diagonal. With the trace it gives det(A − λI).
det A = 4 × 3 − 1 × 2
det([[4
−
λ,
1],
[2,
3
−
λ]])
=
(4
−
λ)(3
−
λ)
−
1
×
2
=
λ
2
−
7λ
+
10
det A = 10
Step 3 · Find discriminant tr² − 4 det (Δ)
Use Δ = (tr A)² − 4 × (det A)
The
discriminant
of
λ
2
−
(tr
A)λ
+
det
A
=
0
says
whether
its
roots
are
real.
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
on
λ
2
−
(tr
A)λ
+
det
A
=
0,
with
the
+
sign
for
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 same formula with the − sign gives the smaller root.
λ₂
=
7 −
√
9
2
λ₂
=
7 − 3
2
λ₂
=
4
2
λ₂
=
2
Step 6 · Find slope of v₁ (p₁)
Use p₁ = (λ₁ − a) ÷ b
Row
1
of
(A
−
λI)v
=
0
reads
(
a
−
λ)x
+
by
=
0.
Put
x
=
1
and
solve
for
y.
p₁
=
5 − 4
1
Check: [[4, 1], [2, 3]] × ⟨1, 1⟩ = ⟨5, 5⟩ = 5 × ⟨1, 1⟩
p₁
=
1
Step 7 · Find slope of v₂ (p₂)
Use p₂ = (λ₂ − a) ÷ b
Row
1
of
(A
−
λI)v
=
0
reads
(
a
−
λ)x
+
by
=
0.
Put
x
=
1
and
solve
for
y.
p₂
=
2 − 4
1
Check: [[4, 1], [2, 3]] × ⟨1, −2⟩ = ⟨2, −4⟩ = 2 × ⟨1, −2⟩
p₂
=
−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
✓
1
=
5 − 4
1
✓
−2
=
2 − 4
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
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