Skip to main content

Section 36.3 Terminology and notation

pairing

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

Remark 36.3.1.

We might say real pairing if we want to specify a pairing that outputs real numbers, or complex pairing if we want to specify a pairing that outputs complex numbers.

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 \(\uvec{z}\)

As we will discuss in Section 36.4, an inner product will allow us to transplant geometric notions of vectors from \(\R^n\) into other spaces.

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*}

Finally, here are some definitions related to constructing alternative inner products on \(\R^n\) and \(\C^n\text{.}\)

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 \(\C^n\)