Understanding Matrix Multiplication

Matrix Multiplication

• Matrices $A$ and $B$ can be multiplied if the number of columns in $A$ equals the number of rows in $B$.
• If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then their product $C = A \cdot B$ is an $m \times p$ matrix.
• The element $c_{ik}$ of the product matrix $C$ is calculated as $c_{ik} = \sum_{j=1}^{n} a_{ij}b_{jk}$.
• Matrix multiplication is not commutative; in general, $A \cdot B \neq B \cdot A$.
• Matrix multiplication is associative: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$.
• The identity matrix, denoted as $I_n$, is a square matrix with 1s on the main diagonal and 0s elsewhere.
• For any matrix $A \in K^{m \times n}$, $A \cdot I_n = A = I_m \cdot A$.

Complex Number Comparison

Algebraic Form

• Any complex number $z$ can be uniquely written as $z = a + ib$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).
• $a$ is the real part of $z$, denoted as $\Re(z)$.
• $b$ is the imaginary part of $z$, denoted as $\Im(z)$.
• Two complex numbers, $z = a + ib$ and $z' = a' + ib'$, are equal if and only if $a = a'$ and $b = b'$.

Conjugate

• The conjugate of a complex number $z = a + ib$ is $\overline{z} = a - ib$.
• Properties of conjugates:
  • $\overline{z + z'} = \overline{z} + \overline{z'}$
  • $\overline{z \times z'} = \overline{z} \times \overline{z'}$
  • $\overline{\left( \frac{z}{z'} \right)} = \frac{\overline{z}}{\overline{z'}}$
  • $\overline{\overline{z}} = z$
  • $z + \overline{z} = 2\Re(z)$
  • $z - \overline{z} = 2i\Im(z)$
  • $z \in \mathbb{R} \Leftrightarrow z = \overline{z}$
  • $z \in i\mathbb{R} \Leftrightarrow z = -\overline{z}$

Module

• The modulus of a complex number $z = a + ib$ is $|z| = \sqrt{a^2 + b^2}$.
• Properties of modulus:
  • $|z| = |\overline{z}|$
  • $|z| = 0 \Leftrightarrow z = 0$
  • $|zz'| = |z||z'|$
  • $\left| \frac{z}{z'} \right| = \frac{|z|}{|z'|}$, if $z' \neq 0$
  • $|z^n| = |z|^n$, for any natural number $n$
  • $|z + z'| \leq |z| + |z'|$ (triangle inequality)

Trigonometric Form

• Any non-zero complex number $z$ can be associated with an argument $\theta$, unique up to $2\pi$, such that $a = |z| \cos(\theta)$ and $b = |z| \sin(\theta)$.
• $\theta = \arg(z)$.
• The trigonometric form of $z$ is $z = |z|(\cos(\theta) + i\sin(\theta))$.
• Properties of arguments:
  • $\arg(zz') = \arg(z) + \arg(z') \pmod{2\pi}$
  • $\arg\left(\frac{z}{z'}\right) = \arg(z) - \arg(z') \pmod{2\pi}$
  • $\arg(z^n) = n\arg(z) \pmod{2\pi}$, for any natural number $n$
  • $\arg(\overline{z}) = -\arg(z) \pmod{2\pi}$
• Moivre's Formula: $(\cos(\theta) + i\sin(\theta))^n = \cos(n\theta) + i\sin(n\theta)$
• Euler's Formula:
  • $\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}$
  • $\sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

Exponential Form

• For any real number $\theta$, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$.
• Any non-zero complex number $z$ can be written as $z = |z|e^{i\theta}$, where $\theta = \arg(z)$.
• Properties of exponential form:
  • $e^{i(\theta + 2\pi)} = e^{i\theta}$
  • $e^{i\theta}e^{i\theta'} = e^{i(\theta + \theta')}$
  • $\frac{e^{i\theta}}{e^{i\theta'}} = e^{i(\theta - \theta')}$
  • $(e^{i\theta})^n = e^{in\theta}$, for any natural number $n$
  • $\overline{e^{i\theta}} = e^{-i\theta}$