Complex Flashcards
Definition: Complex Number
z = a + bi
Definition of Complex Inverse
Z^-1 = (a-bi)/a^2+b^2
Definition: Complex Exponential
e^iθ = cosθ + isinθ
A property of complex exponential e multiplication
e^iθ1*e^iθ2=e^i(θ1+θ2)
Definition: Complex Conjugate (z=a+bi)
z^∗ = a − bi
Theorem: Properties of Complex Conjugation
(i) (z + w)^∗ = z^∗ + w^∗
(ii) (z-w)^* = z^* -w^*
(iii) (zw)^* =z^w^
(iv) (1/z)^* = 1/(z^*) (with z != 0)
Definition: Modulus of a Complex Number
|z| =√(zz^∗) =
√(Re(z)^2 + Im(z)^2)
Theorem: Facts about the Modulus
(i) |z| ∈ R
(ii) |z| ≥ 0
(iii) |z|^2 = zz^∗
(iv) z = a + bi with a, b ∈ R =⇒ |z| =√(a^2 + b^2)
(v) If z is pure real, then |z| is just the absolute value of z.
Definition: Phase (z is phase)
|z| = 1
Definition: Complex Conjugation of a Matrix
(A^∗)ij = (Aij)^*
Theorem: Properties of Matrix Conjugation
(i) (A + B)^∗ = A^∗ + B^∗
(ii) (AB)^∗ = A^∗B^∗
(iii) (λA)^∗ = λ^∗A^∗
(iv) (A^∗)^∗ = A
Definition: Adjoint/Dagger of a Matrix
(A†)ij = (Aji)^*
Theorem: Properties of the adjoint
(i) (A + B)† = A† + B†
(ii) (AB)† = B†A†
(iii) (λA)† = λ^∗A†
(iv) (A†)† = A
Definition: Hermitian Matrix
A = A†
Special Type of Matrix
Theorem: Complex Hermitian Matrices if symmetric as well or Complex Unitary if also orthogonal
A is both symmetric and hermitian =⇒ A is Real
A is both orthogonal and unitary =⇒ A is Real
Definition: Unitary matrix
A^−1 = A†
A†A=I^(n)=AA†
Theorem: Fact about Real Unitary Matrices
A is orthogonal ⇐⇒ A is unitary
