6.4 Hermitian Matrices

Ch 6: Eigenvalues
6.4 Hermitian Matrices
We consider matrices with complex entries (a i,j ∈ C) versus real entries (a i,j ∈ R).
1. in R the length of a real number x is |x| = the length from the origin to the number
(either in the positive or the negative direction)
2. in C the length of a complex number z = a + bi is |z| = √ a 2 + b 2 = the length of the
vector [ a b ] (from the origin to the point (a, b)). Also z 2 = a 2 + b 2 .
3. in C the scalars are complex numbers z, addition of complex numbers: (a + bi) +
(c + di) = (a + c) + (b + d)i and product of complex numbers: (a + bi)(c + di) =
(ac − bd) + (ad + bc)i, and C with + and · is a vector space.
4. Particularly C is a normed vector space with the vectors z = (z 1 , z 2 , . . . , z n ) T , and
the norm ||z|| = √ (¯z T z) = √ ¯z 1 z 1 + ¯z 2 z 2 + . . . + z¯
n z n , which is the square root of the
inner product, thus a real number. Here ¯z = ( ¯z 1 , ¯z 2 , . . . , z¯
n ) T is the conjugate of z.
We denote by z H = ¯z T , and so ||z|| = √ z H z
5. the inner product of z and w is the complex number 〈z, w〉 = w H z
6. if z is a vector in the complex vector space with the orthonormal basis {w 1 , w 2 . . . , w n },
then we can write z as a linear combination of the vector basis as z = ∑ n
i=1 〈z, w i〉w i
(see Exercise 2 of Homework)
7. if A is a matrix in C m×n , then A H is the matrix whose every entry is the conjugate
of the corresponding entry of A.
⎡
⎤
⎡
⎤T
⎡
⎤
2 − i 1 −2i
2 + i 1 +2i 2 + i −5i 0
For example, if A = ⎣ 5i 0 5 − i⎦ then A H = ⎣ −5i 0 5 + i⎦
= ⎣ 1 0 5 ⎦
0 5 −5i
0 5 5i 2i 5 5i
8. vectors in R n versus in C n and matrices in R n versus in C n :
R n
C n
〈x, y〉 = y T x
〈z, w〉 = w H z
〈x, x〉 ≥ 0 with equality iff x = 0 〈z, z〉 ≥ 0 with equality iff z = 0
〈x, y〉 = 〈y, x〉 〈z, w〉 = 〈w, z〉
〈αx + βy, z〉 = α〈x, z〉 + β〈y, z〉 〈αz + βu, w〉 = α〈z, w〉 + β〈u, w〉
||x|| 2 = 〈x, x〉 = x T x
||z|| 2 = 〈z, z〉 = z H z
R n×n
C n×n
(A T ) T = A (A H ) H = A
(αA + βB) T = αA T + βA T (α, β ∈ R) (αA + βB) H = ᾱA H + ¯βB H (α, β ∈ C)
(AB) T = B T A T
(AB) H = B H A H
if A T = A ⇐⇒ A = symmetric if A H = A ⇐⇒ A = Hermitian
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9. A matrix A is a Hermitian matrix if A H = A (they are ideal matrices in C since
properties that one
[
would expect
]
for matrices will probably
[
hold).
]
1 2 − i
1 2 + i
For example A =
is Hermitian since
2 + i 0
Ā = and so
2 − i 0
[ ]
A H 1 2 − i
= ĀT =
= A
2 + i 0
10. if A is Hermitian, then[ A is symmetric. ] However the converse fails, and here is a
1 2 − i
counterexample: A =
. However if A ∈ R
2 − i 0
n×n is symmetric, then it is
Hermitian.
Symmetric and orthogonal matrices in R n×n
Hermitian and unitary matrices in C n×n
Defn: if A T = A ⇐⇒ A = symmetric
Defn: if A H = A ⇐⇒ A = Hermitian
A = symmetric =⇒ A is a square matrix
A = Hermitian =⇒ A is a square matrix
a pure complex matrix cannot be Hermitian
(the diagonal must have real entries)
A = symmetric =⇒ λ i ∈ R, ∀i
A = Hermitian =⇒ λ i ∈ R, ∀i
A = symmetric =⇒ eigenvectors
A = Hermitian =⇒ eigenvectors
belonging to distinct eigenvalues are orthogonal belonging to distinct eigenvalues are orthogonal
(see #5 page 353)
Q is orthogonal if its column vectors
U is unitary if its column vectors
form an orthonormal set
form an orthonormal set
(i.e. Q T Q = I = QQ T ) (i.e. U H U = I = UU H )
Q is orthogonal =⇒ Q −1 = Q T
U is unitary =⇒ U −1 = U H
if the eigenvalues of matrix A are all distinct, if A is an Hermitian matrix A,
(or algebraic multipl λ i = geom multipl λ i , ∀i) =⇒ ∃U = unitary and it diagonalizes A
=⇒ ∃X = nonsingular and it diagonalizes A (i.e. the diagonal matrix T is
(i.e. the diagonal matrix D is T = U H AU or A = UT U H )
D = X −1 AX or A = XDX −1 ) T is first shown to be upper triangular in Thm 6.4.3
(see Remark 3 page 308) and then that it is shown to be diagonal in Thm 6.4.4
11. These three give the last row in the table above:
(a) if the eigenvalues of an Hermitian matrix A are all distinct, then ∃U that is
unitary and it diagonalizes A. In this case U has as columns the normalized
eigenvectors of A
(b) Schur’s Theorem: If A is n × n, then ∃U a unitary matrix such that T = U H AU
is upper triangular matrix.
(c) Spectral Theorem: If A is Hermitian, then ∃U a unitary matrix such that U H AU
is a diagonal matrix.
Note that if some eigenvalue λ j has algebraic multiplicity ≥ 2, then the eigenvectors
corresponding to λ j are not orthonormal, and so we use Gram-Schmidt
to normalize them (we use Gram Schmidt for each set of eigenvectors that correspond
to each repeated eigenvalue)
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R n
C n
〈x, y〉 = y T x
〈z, w〉 = w H z
〈x, x〉 ≥ 0 with equality iff x = 0 〈z, z〉 ≥ 0 with equality iff z = 0
〈x, y〉 = 〈y, x〉 〈z, w〉 = 〈w, z〉
〈αx + βy, z〉 = α〈x, z〉 + β〈y, z〉 〈αz + βu, w〉 = α〈z, w〉 + β〈u, w〉
||x|| 2 = 〈x, x〉 = x T x
||z|| 2 = 〈z, z〉 = z H z
R n×n
C n×n
(A T ) T = A (A H ) H = A
(αA + βB) T = αA T + βA T (α, β ∈ R) (αA + βB) H = ᾱA H + ¯βB H (α, β ∈ C)
(AB) T = B T A T
(AB) H = B H A H
if A T = A ⇐⇒ A = symmetric if A H = A ⇐⇒ A = Hermitian
Symmetric and orthogonal matrices in R n×n
Hermitian and unitary matrices in C n×n
Defn: if A T = A ⇐⇒ A = symmetric
Defn: if A H = A ⇐⇒ A = Hermitian
A = symmetric =⇒ A is a square matrix
A = Hermitian =⇒ A is a square matrix
a pure complex matrix cannot be Hermitian
(the diagonal must have real entries)
A = symmetric =⇒ λ i ∈ R, ∀i
A = Hermitian =⇒ λ i ∈ R, ∀i
A = symmetric =⇒ eigenvectors
A = Hermitian =⇒ eigenvectors
belonging to distinct eigenvalues are orthogonal belonging to distinct eigenvalues are orthogonal
(see #5 page 353)
Q is orthogonal if its column vectors
U is unitary if its column vectors
form an orthonormal set
form an orthonormal set
(i.e. Q T Q = I = QQ T ) (i.e. U H U = I = UU H )
Q is orthogonal =⇒ Q −1 = Q T
U is unitary =⇒ U −1 = U H
if the eigenvalues of matrix A are all distinct, if A is an Hermitian matrix A,
(or algebraic multipl λ i = geom multipl λ i , ∀i) =⇒ ∃U = unitary and it diagonalizes A
=⇒ ∃X = nonsingular and it diagonalizes A (i.e. the diagonal matrix T is
(i.e. the diagonal matrix D is T = U H AU or A = UT U H )
D = X −1 AX or A = XDX −1 ) T is first shown to be upper triangular in Thm 6.4.3
(see Remark 3 page 308) and then that it is shown to be diagonal in Thm 6.4.4
3