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 \(\C^n\)