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제 22강. 복소행렬과 에르미트행렬

Complex Matrix

Complex Number : $a + i b$ $(i^{2} = - 1)$
Complex conjugate : $(a + i b)^{*} = a - i b$
(a : real-number / b : imaginary part)
$a = r \cos θ$ / $b = r \sin θ$
$Z = a + i b$
$= r (\cos θ + i \sin θ)$
$= r e^{i θ}$
=> Euler Formula

length

$| | x^{2} | | = | x_{1} | + | x_{2} | + \dots + | x_{n}^{2} |$
$= {x_{1}}^{} x_{1} + {x_{2}}^{} x_{2} + \dots + {x_{n}}^{} x_{n}$
$= x^{ T} x$

Inner product

$x^{* T} y (\neq y^{* T} x)$
$= x^{H} y (x^{H} = x^{* T}$ ( Hermitian ) ( conjugate transpose )
(1) Orthogonal $x$ and $y$ = $x^{H} y = 0$
(2) $| | x | |^{2} = x^{H} x = | x_{1} |^{2} + | x_{2} |^{2} + \dots + | x_{n} |^{2}$
(3) $(A B)^{H} = B^{H} A^{H}$
(4) $(x^{H} y)^{H} = y^{H} x$
$x^{H} y < - > y^{H} x$ "conjugate"

Hermitian Matrices

Real components : $A^{T} = A$ => "Symmetric"
Complex : $A^{H} = A$ => "Hermitian"
$a_{i j} = a {j i}^{*} (i \neq j)$
$a_{i} j (i = j)$ (=real)

Symmetric matrices $\subset$ Hermitian matrices

Properties of Hermitian or Symmetric Matrix

(1) $X^{H} A X$ (quadratic form) is "real"

$R = A^{H} A$ =>
$X^{H} R X = X^{H} A^{H} A X$
$= (A X)^{H} A X$
$= | | A X | |^{2} > 0$

(2) If $A^{H} = A$ , every eigenvalue is real

$A x = λ x$ for complex vecctor x
각 항에 $x^{H}$ 를 곱하면
$x^{H} A x = x^{H} λ x = λ x^{H} x = λ | | x | |^{2}$
$λ = \frac{x^{H} A x}{| | x | |^{2}}$
분자, 분모 모두 real(실수)라서 람다도 real 이다.

(3) Eigenvectors are orthogonal

$A x_{1} = λ_{1} x_{1}, A x_{2} = λ_{2} x_{2}, (λ_{1} \neq λ_{2} a n d A^{H} = A)$
$(A x_{1})^{H} x_{2} = x_{1}^{H} A^{H} x_{2} = x_{1}^{H} A x_{2}$
$= (λ_{1} x_{1})^{H} x_{2} = x_{1}^{H} λ_{1}^{H} x_{2} = λ_{1} x_{1}^{H} x_{2}$
$x_{1}^{H} A x_{2} = x_{1}^{H} λ_{2} x_{2} = λ_{2} x_{1}^{H} x_{2} = λ_{1} x_{1}^{H} x_{2}$
$= (λ_{2} - λ_{1}) x_{1}^{H} x_{2} = 0 (λ_{1} \neq λ_{2})$
그래서
$x_{1}^{H} x_{2} = 0$ or $x_{2}^{H} x_{1} = 0$ 이어서 수직이게 되는 것이다.
Angle length preserved
$A = S Λ S^{- 1} = Q Λ Q^{- 1} = Q Λ Q^{T}$
$(Q x)^{T} (Q y) = x^{T} Q^{T} Q y = x^{T} y$
$| | Q x | |^{2} = (Q x)^{T} (Q x) = X^{T} Q^{T} Q x = | | x | |^{2}$

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[KOCW 선형대수] 22강. 복소행렬과 에르미트행렬

제 22강. 복소행렬과 에르미트행렬

Complex Matrix

Complex Number : $a + i b$ $(i^{2} = - 1)$
Complex conjugate : $(a + i b)^{*} = a - i b$
(a : real-number / b : imaginary part)
$a = r \cos θ$ / $b = r \sin θ$
$Z = a + i b$
$= r (\cos θ + i \sin θ)$
$= r e^{i θ}$
=> Euler Formula

length

$| | x^{2} | | = | x_{1} | + | x_{2} | + \dots + | x_{n}^{2} |$
$= {x_{1}}^{} x_{1} + {x_{2}}^{} x_{2} + \dots + {x_{n}}^{} x_{n}$
$= x^{ T} x$

Inner product

Hermitian Matrices

Real components : $A^{T} = A$ => "Symmetric"
Complex : $A^{H} = A$ => "Hermitian"
$a_{i j} = a {j i}^{*} (i \neq j)$
$a_{i} j (i = j)$ (=real)

Symmetric matrices $\subset$ Hermitian matrices

Properties of Hermitian or Symmetric Matrix

(1) $X^{H} A X$ (quadratic form) is "real"

$R = A^{H} A$ =>
$X^{H} R X = X^{H} A^{H} A X$
$= (A X)^{H} A X$
$= | | A X | |^{2} > 0$

(2) If $A^{H} = A$ , every eigenvalue is real

$A x = λ x$ for complex vecctor x
각 항에 $x^{H}$ 를 곱하면
$x^{H} A x = x^{H} λ x = λ x^{H} x = λ | | x | |^{2}$
$λ = \frac{x^{H} A x}{| | x | |^{2}}$
분자, 분모 모두 real(실수)라서 람다도 real 이다.

(3) Eigenvectors are orthogonal

티스토리툴바

제 22강. 복소행렬과 에르미트행렬

Complex Matrix

Complex Number : 𝑎+𝑖𝑏 (𝑖2=−1)Complex conjugate : (𝑎+𝑖𝑏)∗=𝑎−𝑖𝑏(a : real-number / b : imaginary part)𝑎=𝑟cos𝜃 / 𝑏=𝑟sin𝜃𝑍=𝑎+𝑖𝑏Z=a+ib=𝑟(cos𝜃+𝑖sin𝜃)=𝑟𝑒𝑖𝜃=> Euler Formula

length

||𝑥2||=|𝑥1|+|𝑥2|+⋯+|𝑥2𝑛|=𝑥1∗𝑥1+𝑥2∗𝑥2+⋯+𝑥𝑛∗𝑥𝑛=𝑥∗𝑇𝑥

Inner product

Hermitian Matrices

Real components : 𝐴𝑇=𝐴 => "Symmetric"Complex : 𝐴𝐻=𝐴 => "Hermitian"𝑎𝑖𝑗=𝑎𝑗𝑖∗(𝑖≠𝑗)𝑎𝑖𝑗(𝑖=𝑗) (=real)

Symmetric matrices ⊂ Hermitian matrices

Properties of Hermitian or Symmetric Matrix

(1) 𝑋𝐻𝐴𝑋 (quadratic form) is "real"

𝑅=𝐴𝐻𝐴 =>𝑋𝐻𝑅𝑋=𝑋𝐻𝐴𝐻𝐴𝑋=(𝐴𝑋)𝐻𝐴𝑋=||𝐴𝑋||2>0

(2) If 𝐴𝐻=𝐴, every eigenvalue is real

𝐴𝑥=𝜆𝑥 for complex vecctor x각 항에 𝑥𝐻를 곱하면𝑥𝐻𝐴𝑥=𝑥𝐻𝜆𝑥=𝜆𝑥𝐻𝑥=𝜆||𝑥||2𝜆=𝑥𝐻𝐴𝑥||𝑥||2분자, 분모 모두 real(실수)라서 람다도 real 이다.

(3) Eigenvectors are orthogonal

티스토리툴바

Complex Number : $a + i b$ $(i^{2} = - 1)$
Complex conjugate : $(a + i b)^{*} = a - i b$
(a : real-number / b : imaginary part)
$a = r \cos θ$ / $b = r \sin θ$
$Z = a + i b$
$= r (\cos θ + i \sin θ)$
$= r e^{i θ}$
=> Euler Formula

$| | x^{2} | | = | x_{1} | + | x_{2} | + \dots + | x_{n}^{2} |$
$= {x_{1}}^{} x_{1} + {x_{2}}^{} x_{2} + \dots + {x_{n}}^{} x_{n}$
$= x^{ T} x$

Real components : $A^{T} = A$ => "Symmetric"
Complex : $A^{H} = A$ => "Hermitian"
$a_{i j} = a {j i}^{*} (i \neq j)$
$a_{i} j (i = j)$ (=real)

Symmetric matrices $\subset$ Hermitian matrices

(1) $X^{H} A X$ (quadratic form) is "real"

$R = A^{H} A$ =>
$X^{H} R X = X^{H} A^{H} A X$
$= (A X)^{H} A X$
$= | | A X | |^{2} > 0$

(2) If $A^{H} = A$ , every eigenvalue is real

$A x = λ x$ for complex vecctor x
각 항에 $x^{H}$ 를 곱하면
$x^{H} A x = x^{H} λ x = λ x^{H} x = λ | | x | |^{2}$
$λ = \frac{x^{H} A x}{| | x | |^{2}}$
분자, 분모 모두 real(실수)라서 람다도 real 이다.