Complex Numbers for Linear Algebra

Definition The imaginary unit $i$ is a number with $i^2 = -1$. A complex number is an expression $z = a + bi$ with real numbers $a$ and $b$. Its real part is $\operatorname{Re} z = a$ and its imaginary part is the real number $\operatorname{Im} z = b$. The set of all complex numbers is written $\mathbb{C}$. Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal. A real number $a$ is the complex number $a + 0i$, so $\mathbb{R}$ sits inside $\mathbb{C}$. Sums, differences and products follow the usual rules of algebra together with $i^2 = -1$,

The square of a real number is never negative, so $i$ is not a real number. Its powers $i, i^2, i^3, i^4, \dots$ are $i, -1, -i, 1$ and then repeat in the same cycle of four.

Definition The complex plane represents the complex number $a + bi$ by the point $(a, b)$ of the coordinate plane. The horizontal axis, made of the points $(a, 0)$, is the real axis, and the vertical axis, made of the points $(0, b)$, is the imaginary axis. Each complex number corresponds to exactly one point and each point to exactly one complex number. Addition of complex numbers is addition of coordinate pairs, so a sum can be drawn with the tip-to-tail rule for vectors. Multiplication is not done coordinate by coordinate, as the product rule shows.

Definition For every complex number $a + bi$,

so multiplication by $i$ sends the point $(a, b)$ to $(-b, a)$. Treating points as column vectors, with a tuple standing for a column, this is the linear map of $\mathbb{R}^2$ with matrix

It is the counterclockwise rotation of the plane about the origin through a right angle. For a nonzero point $(a, b)$ the image has the same distance $\sqrt{a^2 + b^2}$ from the origin, it is perpendicular to $(a, b)$ because $a(-b) + ba = 0$, and the turn goes counterclockwise because the matrix with columns $(a, b)$ and $(-b, a)$ has the positive determinant $a^2 + b^2$. Multiplying by $i$ twice multiplies by $i^2 = -1$, a half turn that sends $(a, b)$ to $(-a, -b)$, and four times multiplies by $i^4 = 1$. Multiplication by $-i$ is the clockwise quarter-turn.

Definition The complex conjugate of $z = a + bi$ is $\overline{z} = a - bi$. In the complex plane it is the reflection across the real axis. For all complex numbers $z$ and $w$,

and $\overline{\overline{z}} = z$. A complex number is real exactly when it equals its conjugate. The sum $z + \overline{z} = 2a$ is real, and the product of a number with its conjugate is a nonnegative real number,

Definition The modulus of $z = a + bi$ is the real number

the distance from $0$ to the point $(a, b)$ in the complex plane. It satisfies $\lvert z \rvert \ge 0$, with equality only for $z = 0$, and for a real number it is the usual absolute value. Also $\lvert \overline{z} \rvert = \lvert z \rvert$. The modulus of a product is the product of the moduli, $\lvert zw \rvert = \lvert z \rvert \lvert w \rvert$, because

and both moduli are nonnegative.

Definition Let $w = c + di$ be a nonzero complex number, so $c$ and $d$ are not both zero. For every complex number $z$, the quotient $z / w$ is the unique complex number $q$ with $qw = z$. Multiplying the numerator and the denominator by $\overline{w}$ turns the denominator into the positive real number $\lvert w \rvert^2 = c^2 + d^2$, so

In particular the reciprocal of $w$ is

The number $0$ has no reciprocal, since $0 \cdot q = 0$ for every $q$, so division by $0$ stays undefined.