Complex Vector Space

A $\mathbf z$ is a complex vector (denoted by $\mathbf z \in \mathbb C^n$)


Norm

How do we define the length of a complex vector?

  • $\| \mathbf z \|^2 = \langle \mathbf z, \mathbf z \rangle = \mathbf z^T \mathbf z$ is no good:
    • length should be positive
    • consider, for example, vector $(1, i)$
    • $\| (1, i) \|^2$ would be $1^2 + i^2 = 0$
  • what we really want is $\langle \mathbf z, \mathbf z \rangle = \overline {\mathbf z}^T \mathbf z$
    • where $\overline {\mathbf z}$ is a Complex Conjugate, i.e. $\overline {\mathbf z} = (\overline z_1, \ ... \ , \overline z_n)$
    • this way each component of $\langle \mathbf z, \mathbf z \rangle$ contributes a strictly positive number to the overall dot product
    • so $\| (1, i) \|^2$ is $1 - i^2 = 2$
    • thus, $\| (1, i) \| = \sqrt{2}$


Hermitian

The way to transpose and take the conjugate at the same time

  • $\mathbf z^H$ is $\overline {\mathbf z}^T$
  • so we say $\| \mathbf z \|^2 = \mathbf z^H \mathbf z = \sum | z_i |^2$
  • hermitian operator also applies to matrices
  • $A^H$ is $\overline A^T$


Inner Product

The same for the dot product

  • $\langle x, y \rangle$ is not $\mathbf x^T \mathbf y$
  • it's $\langle x, y \rangle = \mathbf x^H \mathbf y$


Symmetric Matrices

What about symmetric matrices in $\mathbb C^{n \times n}$?

  • The definition that $A$ is symmetric if $A^T = A$ is for $\mathbb R$, not $\mathbb C$
  • the complex version of symmetry is $\overline {A}^T = A$, or $A^H = A$ 0 using the Hermitian operator
  • note that diagonal of a symmetric matrix must be real, because otherwise real values are complex conjugates of each others


Unitary Matrices

Can a complex matrix be orthogonal?

  • $Q^T Q = I$ is orthogonal matrix for $\mathbb R$
  • what about $\mathbb C$?
  • yes, it's possible: $Q^H Q = I$ and all it's columns $\mathbf q_1, \ ... \ , \mathbf q_n$ are orthonormal
  • vectors $\mathbf q_1, \ ... \ , \mathbf q_n$ are orthonormal when $\mathbf q_i^H \mathbf q_j = \begin{cases}

1 & \text{ if } i = j \\ 0 & \text{ if } i \ne j \end{cases}$

  • but here instead of "orthogonal" they are usually called "unitary" matrices


Sources