A complex number is of the form \(z=a+ib\), where \(a, b\in \mathbb{R}\) and \(i=\sqrt{-1}\) is the imaginary unit. The set of complex numbers is \(\mathbb{C}\).
For every complex number \(z\), it’s modulus \(\vert z\vert = \sqrt{a^2+b^2}\) and \(\arg (z) = \tan^{-1}\dfrac{b}{a}\).
For \(z=a+ib\), the complex conjugate is obtained by replacing \(i\) with \(-i\), that is \(\bar{z} =a-i b\).
Complex vector: An \(n\)-dimensional complex vector is an \(n\times 1\) matrix whose entries are complex numbers: \[x = [x_1\ x_2\ \ldots \ x_n]^T \quad \text{ for } x_i \in \mathbb{C}\] This complex vector \(x\in \mathbb{C}^n\).
Inner product: For complex vectors \(x, y \in \mathbb{C}^n\), the inner product of \(x\) and \(y\) is defined as \(\langle x, y \rangle = \bar{x}^T y\). Notice, that it is not commutative. Infact, \(\langle y, x \rangle = \overline{ \langle x, y \rangle}\).
Norm of a complex vector: For the vector \[x = [x_1\ x_2\ \ldots \ x_n]^T \in \mathbb{C}^n,\] the norm is defined as \[\Vert x \Vert = \sqrt { \langle x, x\rangle}\] which gives the working rule as \[\Vert x \Vert^2 = \vert x_1 \vert^2 + \vert x_2 \vert^2 + \ldots + \vert x_n \vert^2\]
For a square complex matrix \(A\), its conjugate transpose, denoted by \(A^*\) is obtained by taking the complex conjugate of every element and then transposing the matrix.
A square complex matrix \(A\) is said to be Hermtian, if and only if \(A^* = A\). (The corresponding real analogue would be symmetric matrix.)
In a Hermitian matrix, the diagonal entries are purely real and positions that are symmetrical with respect to the main diagonal contain entires that are complex conjugate of each other.
All eigenvalues of a Hermitian matix are real.
Eigenvectors corresponding to distinct eigenvalues are orthogonal.
A square complex matrix \(U\) is said to be unitary, if and only if \(U^*U = I\) or \(U^* = U^{-1}\) (The corresponding real analogue would be orthogonal matrix.)
The columns of a unitary matrix are orthonormal.
Unitary matrices preserve lengths: For any vextor \(x\) and a unitary matrix \(U\): \[\Vert Ux \Vert = \Vert x \Vert\]
Eigenvalues of a unitary matrix have absolute value \(1\), that is if \(\lambda\) be an eigenvalue of a unitary martrix, then \(\vert \lambda \vert = 1\).
Eigenvectors of a unitary matrix, corresponding to distinct eigenvalues are orthogonal.
Schur’s theorem: Any \(n \times n\) matrix \(A\) is similar to an upper triangular matrix, i.e. \(\exists\) an upper triangular matrix \(T\) and a unitary matrix \(U\) so that \(A = UTU^*\).
Spectral theorem: A Hermitian matrix is unitarily diagoinalizable. If \(A\) is Hermitian, \(\exists\) a unitray matrix \(U\) such that \(A=U\Lambda U^*\) for a diagonal matrix \(\Lambda\). The matrix \(U\) contains, as its columns, the normalized eigenvectors of \(U\), and \(\Lambda\) contains, along the main diagonal, the corresponding eigenvalues.