SME
Linear Algebra · Problem type
Complex eigenvalues: a turn and a stretch
Use this for “Find the eigenvalues of [[1, −1], [1, 1]] and the angle and scale factor of the map.”
Diagram
|z|
=
√
1
2
+
1
2
=
√
2
≈
1.41
arg z = 45°.
The conjugate z̄ = 1 − i: z reflected across the real axis.
This is an example. Type your own values to start a new problem.
A
=
λ
=
a
±
b
i
=
r
(cos
θ (°)
±
i
sin
θ (°)
)
tr A
Trace of A
worked out (not typed)
det A
Determinant of A
worked out (not typed)
Formulas
c
=
−
b
−1 = −1
tr
A
=
2
×
a
2 = 2 × 1
det
A
=
a
2
+
b
2
2
=
1
2
+
1
2
r
=
√
a
2
+
b
2
1.4142
=
√
1
2
+
1
2
θ
=
arg(
a
+
bi)
45 = arg(1 + 1i)
Assumptions
•
det(A − λI) = λ² − 2aλ + a² + b², so λ = a ± bi. With b ≠ 0 no real line is kept.
•
A = r[[cos θ, −sin θ], [sin θ, cos θ]]: it turns the plane by θ and stretches it by r = |λ|.
•
Here b > 0, so θ is from 0° to 180° and the turn is counterclockwise. [[a, b], [−b, a]] turns the other way.
•
The eigenvectors are complex: (1, −i) for a + bi, and (1, i) for a − bi.
Step-by-step
WE KNOW
a = 1, b = 1
FIND
A, row 1, column 2 (c), trace of A (tr A), determinant of A (det A), stretch |λ| (r), turn arg λ (θ)
Step 1 · Find A, row 1, column 2 (c)
Use c = −b
A
turn
and
a
stretch
has
−
b
above
the
diagonal.
c
=
−1
Step 2 · Find trace of A (tr A)
Use tr A = 2 × a
Both
diagonal
entries
are
a
.
tr A = 2 × 1
tr A = 2
Step 3 · Find determinant of A (det A)
Use det A = a² + b²
a
×
a
−
(−
b
)
×
b
.
With
the
trace
it
gives
λ
2
−
(tr
A)λ
+
det
A
=
0,
whose
roots
are
a
±
bi.
det
A
=
1
2
+
1
2
det([[1
−
λ,
−1],
[1,
1
−
λ]])
=
(1
−
λ)(1
−
λ)
−
(−1)
×
1
=
λ
2
−
2λ
+
2
2 ±
√
2
2
− 4 × 2
2
=
1
±
i
det A = 2
Step 4 · Find stretch |λ| (r)
Use r = √(a² + b²)
The
size
of
λ
=
a
+
bi:
it
is
how
much
A
stretches
every
vector.
r
=
√
1
2
+
1
2
r
=
√
1 + 1
r
=
√
2
r
=
1.4142
Step 5 · Find turn arg λ (θ)
Use θ = arg(a + bi)
λ
=
a
+
bi
points
up
and
right,
so
tan⁻¹
gives
its
angle.
θ
=
tan⁻¹(
b
a
)
θ
=
tan⁻¹(
1
1
)
θ
=
45°
Check
Put the numbers back in. Both sides should match.
−1 = −1 ✓
2 = 2 × 1 ✓
2
=
1
2
+
1
2
✓
1.4142
=
√
1
2
+
1
2
✓
tan(45°)
=
1
1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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