REU 2013: Apprentice Program
Summer 2013
Lecture 6: July 5, 2013
Madhur Tulsiani
1
Scribe: David Kim
Eigenvalues & Eigenvectors
Definition 1.1 (Eigenvalue & Eigenvector) Let A ∈ Cn×n . Then λ ∈ C is said to be an
eigenvalue of A if ∃v 6= 0 such that
Av = λv ⇐⇒ (A − λI)v = 0v .
Such a v is an called eigenvector of A with eigenvalue λ.
Exercise 1.2 λ is an eigenvalue of A if and only if det(λI − A) = 0
Exercise 1.3 Let Uλ = {v : Av = λv}. Prove: Uλ is a subspace of Cn .
Definition 1.4 (Characteristic Polynomial) fA (t) = det(tI − A) is called the characteristic
polynomial of A.
From the above, we know that λ is an eigenvalue of A iff lambda is a root of the characteristic
polynomial fA (t).
1 0
. The characteristic polynomial is fA (t) = det(tI − A) =
Example 1.5 Consider A =
0 1
(t − 1)2 . The only eigenvalue is λ = 1, and Uλ = C2 , since Av = Iv = v for all v ∈ C2 .
1 1
Example 1.6 Consider A =
. The characteristic polynomial is still fA (t) = det(tI − A) =
0 1
(t − 1)2 and the only eigenvalue is λ = 1. However, Uλ is now only the one-dimensional space
1
Uλ =
α·
| α∈C .
0
Exercise 1.7 Calculate the eigenvalues of A =
0 1
.
−1 0
Note that the eigenvalue of a real matrix may be complex valued. Also, any polynomial p(t) of
degree n over C can be factored as c(t − λ1 )...(t − λn ) for c, λ1 , ..., λn ∈ C. Thus, the characteristic
polynomial of a matrix always has n (not necessarily distinct) complex roots.
1
Definition 1.8 (Algebraic Multiplicity) The algebraic multiplicity of an eigenvalue λ is the
number of times t − λ appears as a factor in the characteristic polynomial fA (t).
Definition 1.9 (Geometric Multiplicity) The geometric multiplicity of an eigenvalue λ is dim(Uλ ).
Exercise 1.10 Algebraic multiplicity ≥ geometric multiplicity.
1 1
gives an example where the algebraic multiplicity is strictly greater
0 1
than the geometric multiplicity of an eigenvalue.
Note that the matrix
Exercise 1.11 Let Av = λv for v ∈ Cn , A ∈ Rn×n , λ ∈ R. Then Re(v), Im(v) are also eigenvectors
with eigenvalue λ.
Thus, if a real matrix A has an eigenvector with a real eigenvalue λ ∈ R, then it also has a real
eigenvector with the same eigenvalue.
Example 1.12 Calculate
eigenvectors, and the characteristic polynomial for the
the eigenvalues,
cosθ −sinθ
.
rotation matrix, Rθ =
sinθ cosθ
±iθ
fRθ (t) = (t − cosθ)2 + (sinθ)2 = 0 gives
t = cosθ ± isinθ = e . Let λ1 = cosθ + isinθ,
1
i
: α ∈ C}. Note that the
λ2 = cosθ − isinθ. Then Uλ1 = {α
: α ∈ C}, Uλ2 = {α
i
1
eigenvectors do not depend on θ.
Exercise 1.13 If v1 , ..., vn are eigenvectors with distinct eigenvalues λ1 , ..., λn , then v1 , ..., vn are
linearly independent.
Definition 1.14 (Similar Matrices) A and B are similar, A ∼ B, if ∃S ∈ Cn×n such that
A = S −1 BS.
Definition 1.15 (Diagonalizable Matrices) A is diagonalizable if A = S −1 DS where D is a
diagonal matrix. (A is similar to a diagonal matrix).
Exercise 1.16 Prove that similarity between matrices is an equivalence relation.
Exercise 1.17 Prove: If A ∼ B, then fA (t) = fB (t). Thus, if A ∼ B, then they have the same
eigenvalues and each eigenvalue has the same algebraic multiplicity for both A and B.
Exercise 1.18 Let A ∼ B. Then for each λ which is an eigenvalue of A (and hence also of
(A)
(B)
B), show that Uλ is isomorphic to Uλ . Thus, each eigenvalue also has the same geometric
multiplicity for both A and B.
This follows by noting that S −1 BSv = λv ⇒ BSv = λSv. Hence, v 7→ Sv is a bijective linear map
(A)
(B)
from Uλ to Uλ .
2
2
Inner Products and Unitary Matrices
Definition 2.1 (Inner Product) Let u, v ∈ Cn . Then the Hermitian inner product of u and v is
defined as
n
X
hu, vi =
ui vi ,
i=1
where ui denotes the conjugade of ui . Note that this the same as the usual dot-product if u, v ∈ Rn .
P
Note that thepquantity hu, ui = i |ui |2 is always non-negative and is zero only when u = 0. We
define kuk = hu, ui, which extends the notion of length of a vector, to vectors in Cn .
Definition 2.2 (Orthogonal Vectors) Two vectors u, v are said to be orthogonal if hu, vi = 0.
Definition 2.3 (Orthonormal Basis) {v1 , ..., vk } is an orthonormal basis for V if it is a basis
such that
0 if i =
6 j
.
∀i, j hvi , vj i =
1 if i = j
1
1
are orthogonal. After
and
Example 2.4 For the rotation matrix Rθ , the eigenvectors
i
i
1
1
scaling, the vectors to √12 ·
and √12 ·
, we obtain an orthonormal basis of C2 consisting of
i
i
eigenvectors of Rθ .
Definition 2.5 (Unitary Matrix) A matrix U is called a unitary matrix if the columns of U
form an orthonormal basis of Cn .
Definition 2.6 (Adjoint of a matrix) For a matrix A ∈ Mn (C), its adjoint, denoted as A∗ is
T
the matrix defined as (A∗ )ij = aji i.e., A∗ = A .
Note that it follows from the fact that (AB)T = B T AT that we have (AB)∗ = B ∗ A∗ .
Exercise 2.7 U is unitary if and only if U ∗ U = U U ∗ = I.
Note that (U ∗ U )ij = U (i) , U (j) , where U (i) denotes the ith column of U . Hence U ∗ U = I if and
only if the columns form an orthonormal basis. Also, we have that
U ∗U = I ⇔ U U ∗ = I ∗ = I .
Exercise 2.8 Show that If U1 , U2 are unitary, then so is U1 U2 .
3