15.4: Inner Products
- Page ID
- 22938
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Definition: Inner Product
You may have run across inner products, also called dot products, on \(\mathbb{R}^n\) before in some of your math or science courses. If not, we define the inner product as follows, given we have some \(\boldsymbol{x} \in \mathbb{R}^{n}\) and \(\boldsymbol{y} \in \mathbb{R}^{n}\).
Definition: Standard Inner Product
The standard inner product is defined mathematically as:
\[\begin{aligned}
\langle\boldsymbol{x}, \boldsymbol{y}\rangle &=\boldsymbol{y}^{T} \boldsymbol{x} \\
&=\left(\begin{array}{cccc}
y_{0} & y_{1} & \dots & y_{n-1}
\end{array}\right)\left(\begin{array}{c}
x_{0} \\
x_{1} \\
\vdots \\
x_{n-1}
\end{array}\right) \\
&=\sum_{i=0}^{n-1} x_{i} y_{i}
\end{aligned} \nonumber \]
Inner Product in 2-D
If we have\(\boldsymbol{x} \in \mathbb{R}^2\) and \(\boldsymbol{y} \in \mathbb{R}^2\), then we can write the inner product as
\[\langle\boldsymbol{x}, \boldsymbol{y}\rangle=\|\boldsymbol{x}\|\|\boldsymbol{y}\| \cos (\theta) \nonumber \]
where \(\theta\) is the angle between \(\boldsymbol{x}\) and \(\boldsymbol{y}\).
Geometrically, the inner product tells us about the strength of \(\boldsymbol{x}\) in the direction of \(\boldsymbol{y}\).
Example \(\PageIndex{1}\)
For example, if \(\|x\|=1\), then
\[<x, y>=\|y\| \cos (\theta) \nonumber \]
The following characteristics are revealed by the inner product:
- \(\langle \boldsymbol{x}, \boldsymbol{y}\rangle\) measures the length of the projection of \(\boldsymbol{y}\) onto \(\boldsymbol{x}\).
- \(\langle \boldsymbol{x}, \boldsymbol{y}\rangle\) is maximum (for given \(\|\boldsymbol{x}\|\), \(\|\boldsymbol{y}\|\)) when \(\boldsymbol{x}\) and \(\boldsymbol{y}\) are in the same direction (\((\theta=0) \Rightarrow(\cos (\theta)=1)\)).
- \(\langle \boldsymbol{x}, \boldsymbol{y}\rangle\) is zero when \((\cos (\theta)=0) \Rightarrow\left(\theta=90^{\circ}\right)\), i.e. \(\boldsymbol{x}\) and \(\boldsymbol{y}\) are orthogonal.
Inner Product Rules
In general, an inner product on a complex vector space is just a function (taking two vectors and returning a complex number) that satisfies certain rules:
- Conjugate Symmetry: \[\langle \boldsymbol{x}, \boldsymbol{y}\rangle=\overline{\langle \boldsymbol{x}, \boldsymbol{y}\rangle} \nonumber \]
- Scaling: \[\langle\alpha \boldsymbol{x}, \boldsymbol{y} \rangle=\alpha\langle(\boldsymbol{x}, \boldsymbol{y})\rangle \nonumber \]
- Additivity: \[\langle \boldsymbol{x}+\boldsymbol{y}, \boldsymbol{z} \rangle=\langle \boldsymbol{x}, \boldsymbol{z}\rangle+\langle \boldsymbol{y}, \boldsymbol{z}\rangle \nonumber \]
- "Positivity": \[\langle \boldsymbol{x}, \boldsymbol{x} \rangle >0, \boldsymbol{x} \neq 0 \nonumber \]
Definition: Orthogonal
We say that \(\boldsymbol{x}\) and \(\boldsymbol{y}\) are orthogonal if:
\[\langle\boldsymbol{x}, \boldsymbol{y}\rangle=0 \nonumber \]