Discover Linear Algebra

Section 36.3 Terminology and notation

pairing

a function that takes two inputs of the same type and outputs a number

real inner product

a real pairing that satisfies the four Real inner product axioms

real inner product space

a real vector space endowed with a specific real inner product

complex inner product

a complex pairing that satisfies the four Complex inner product axioms

complex inner product space

a complex vector space endowed with a specific complex inner product

complex dot product (of two vectors $\uvec{w}$ and $\uvec{z}$ in $\C^n$)

the quantity

$$\begin{equation*} \udotprod{w}{z} = w_1 \cconj{z}_1 + w_2 \cconj{z}_2 + \dotsb + w_n \cconj{z}_n; \end{equation*}$$

also referred to as the standard inner product of $\uvec{w}$ and $\uvec{z}$

norm (of a vector $\uvec{v}$ in an inner product space)

the quantity $\unorm{v} = \sqrt{\uvecinprod{v}{v}}$

unit vector

a vector whose norm is equal to $1$

angle (between two vectors $\uvec{u}$ and $\uvec{v}$ in a real inner product space)

the angle $\theta$ satisfying both

$$\begin{align*} 0\amp\le\theta\le\pi \amp \amp\text{and} \amp \cos \theta \amp= \frac{\uvecinprod{u}{v}}{\unorm{u}\unorm{v}} \end{align*}$$

positive definite real matrix

a symmetric real matrix $A$ that satisfies $\utrans{\uvec{x}} A \uvec{x} \gt 0$ for all nonzero column vectors $\uvec{x}$ in $\R^n$

positive definite complex matrix

a self-adjoint complex matrix $A$ that satisfies $\adjoint{\uvec{z}} A \uvec{z} \gt 0$ for all nonzero column vectors $\uvec{z}$ in $\C^n$