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To add two complex numbers, add their real parts and imaginary parts separately. Example: $(2 + 3i) + (1 - 4i) = (2 + 1) + (3 - 4)i = 3 - i$.

To multiply two complex numbers, apply the distributive property and remember that $i^2 = -1$. Example: $(1 + 2i)(3 + 4i) = 3 + 4i + 6i + 8i^2 = 3 + 10i - 8 = -5 + 10i$.

The argument of a complex number is the angle $\theta$ that the line representing the number makes with the positive real axis. It's measured in radians. Example: The argument of $1 + \sqrt{3}i$ is $\frac{\pi}{3}$.

A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit satisfying $i^2 = -1$. Example: $3 + 4i$

To subtract two complex numbers, subtract their real and imaginary parts separately. Example: $(5 + 6i) - (2 + 4i) = (5 - 2) + (6 - 4)i = 3 + 2i$.

The absolute value, or modulus, of a complex number $a + bi$ is given by $|a + bi| = \sqrt{a^2 + b^2}$. Example: The modulus of $3 + 4i$ is $\sqrt{3^2 + 4^2} = 5$.

A complex number can be expressed as $re^{i\theta}$, where r is the modulus and $\theta$ is the argument. Example: $5e^{i\pi/3}$ represents the complex number $\frac{5}{2} + \frac{5\sqrt{3}}{2}i$.

The complex conjugate of a complex number $a + bi$ is $a - bi$. It's reflected across the real axis on the complex plane. Example: The conjugate of $3 + 4i$ is $3 - 4i$.

To divide by a complex number, multiply the numerator and denominator by the conjugate of the denominator. Example: $\frac{1 + i}{2 - i} = \frac{(1 + i)(2 + i)}{(2 - i)(2 + i)} = \frac{2 + 3i + i^2}{4 + 2i - 2i - i^2} = \frac{1 + 3i}{4 + 1} = \frac{1}{5} + \frac{3}{5}i$.

De Moivre's Theorem states that $(cos(\theta) + i \sin(\theta))^n = \cos(n\theta) + i \sin(n\theta)$. Example: $(\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))^2 = \cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})$.

Complex matrices can have complex eigenvalues and eigenvectors. The eigenvalue equation is $A\mathbf{v} = \lambda\mathbf{v}$, where $A$ is a matrix, $\lambda$ is an eigenvalue, and $\mathbf{v}$ is an eigenvector. Example: Matrix $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ has eigenvalues $i$ and $-i$.

The inner product of two complex vectors $\vec{u}$ and $\vec{v}$ is denoted by $\langle \vec{u}, \vec{v} \rangle$ and is calculated by taking the conjugate transpose of $\vec{u}$ and multiplying it by $\vec{v}$. Example: If $\vec{u} = \begin{bmatrix}1 + i\\2\end{bmatrix}$ and $\vec{v} = \begin{bmatrix}1\\3i\end{bmatrix}$, $\langle \vec{u}, \vec{v} \rangle = (1 - i)(1) + (2)(3i) = 1 - i + 6i = 1 + 5i$.

To find square roots of complex numbers, express the number in polar form and take the square root of the modulus and half the argument. Example: The roots of $z^2 = -1$ are $z = e^{i(\frac{\pi}{2} + 2k\pi)}$ and $z = e^{i(\frac{3\pi}{2} + 2k\pi)}$ for $k$ integer.

The conjugate transpose (also known as the Hermitian transpose) of a complex matrix is the transpose of the matrix followed by taking the complex conjugate of each entry. Example: The conjugate transpose of $\begin{bmatrix}1 + i & 3\\2 & 4i\end{bmatrix}$ is $\begin{bmatrix}1 - i & 2\\3 & -4i\end{bmatrix}$.

The Cauchy-Schwarz inequality states that for complex vectors $\vec{u}$ and $\vec{v}$, $|\langle \vec{u}, \vec{v} \rangle|^2 \leq |\langle \vec{u}, \vec{u} \rangle| \times |\langle \vec{v}, \vec{v} \rangle|$. Example: For $\vec{u} = \begin{bmatrix}1\\i\end{bmatrix}$ and $\vec{v} = \begin{bmatrix}i\\1\end{bmatrix}$, $|\langle \vec{u}, \vec{v} \rangle|^2 = |(1)(i) + (i)(1)|^2 = |-2|^2 = 4$ and $|\langle \vec{u}, \vec{u} \rangle| \times |\langle \vec{v}, \vec{v} \rangle| = (1 \times 1) \times (1 \times 1) = 1 \times 1 = 1$, hence the inequality holds.

Euler's formula states that for any real number $\theta$, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. Example: $e^{i\pi} + 1 = 0$ illustrates the relationship between $e$, $i$, $\pi$, 1, and 0.

A complex number's polar form is $r(\cos(\theta) + i\sin(\theta))$ where $r$ is the modulus and $\theta$ is the argument. Example: The polar form of $1 + i$ is $\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))$.

To multiply two complex matrices, compute the sum of the products of corresponding entries from the rows of the first matrix and the columns of the second. Example: $\begin{bmatrix}1&-i\\i&1\end{bmatrix} \times \begin{bmatrix}1&2\\2i&3\end{bmatrix} = \begin{bmatrix}1-2i & 1 + 4i\\1+2i & 2+3i\end{bmatrix}$.

Complex matrices are added by adding corresponding entries, just like real matrices. Example: $\begin{bmatrix}1 + i & 2\end{bmatrix} + \begin{bmatrix}3 & 4 - 2i\end{bmatrix} = \begin{bmatrix}4 + i & 6 - 2i\end{bmatrix}$.