MATH 414 Lecture 2

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Inner Product

Most common example is the vector dot product:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \vec{u}, \vec{v} \right\rangle = \vec{u} \cdot \vec{v} = \sum_i u_i \, v_i = \left| \vec{u} \right| \, \left| \vec{v} \right| \, \cos{\theta}}

Generalized Algebraic Structure

We induce a geometry on the algebraic definition

Length: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\| \vec{v} \right\| = \sqrt{\left\langle \vec{v}, \vec{v} \right\rangle}}

Angle Between: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \cos^{-1}{\left( \frac{\left\langle \vec{u}, \vec{v} \right\rangle}{\left\| \vec{u} \right\| \, \left\| \vec{v} \right\|} \right)}}

The most important case is where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = \frac{\pi}{2}} , in which case $\cos {\theta }=0$ and—more importantly—the vectors ${\vec {u}}$ and ${\vec {v}}$ are perperdicular.

Properties of the Inner Product

Positivity: $\left\langle {\vec {u}},{\vec {u}}\right\rangle >0$ unless ${\vec {u}}={\vec {0}}$, in which case $\left\langle {\vec {0}},{\vec {0}}=0\right\rangle$

Conjugate Symmetry: $\left\langle {\vec {u}},{\vec {v}}\right\rangle ={\overline {\left\langle {\vec {v}},{\vec {u}}\right\rangle }}$. In the case of real numbers, the property becomes commutativity: $\left\langle {\vec {u}},{\vec {v}}\right\rangle =\left\langle {\vec {v}},{\vec {u}}\right\rangle$

Homogenuity: $\left\langle c\,{\vec {u}},{\vec {v}}\right\rangle =c\,\left\langle {\vec {u}},{\vec {v}}\right\rangle$

Linearity: $\left\langle {\vec {u}}+{\vec {w}},{\vec {v}}\right\rangle =\left\langle {\vec {u}},{\vec {v}}\right\rangle +\left\langle {\vec {w}},{\vec {v}}\right\rangle$

Real Spaces

for $\mathbb {R} ^{n}$, where ${\vec {x}}={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}$ and ${\vec {y}}={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}$, we have

$\left\langle {\vec {x}},{\vec {y}}\right\rangle =y^{T}\,x=\sum _{k=1}^{n}x_{k}\,y_{k}$

Complex Spaces

for $\mathbb {C} ^{n}$, $\left\langle {\vec {x}},{\vec {y}}\right\rangle$ is given by ${\vec {y}}^{*}\,{\vec {x}}$ (also written ${\vec {y}}^{H}\,{\vec {x}}$), where ${\vec {y}}^{*}={\vec {y}}^{H}$ is the conjugate transpose of ${\vec {y}}$.

Example: ${\vec {y}}={\begin{bmatrix}i\\1\\-1+2i\\3+i\end{bmatrix}}$, ${\vec {y}}^{*}={\begin{bmatrix}-i&1&-1-2i&3-i\end{bmatrix}}$

Some concrete implementations of inner products may not satisfy all properties for all members of the space.

Signal Spaces

Continuous and Discrete

A signal is just $f(t)$ measured on $\left[a,b\right]$.

Signals and Circuits

For current $I(t)$ (essentially our function $f(t)$)

We can measure voltage drop of a signal $V(t)=R\,I(t)$

Power at time $t$ is given by $P(t)=V\,I(t)=R\,I^{2}(t)$

Energy in time interval $\left[a,b\right]$ is given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \int_a^b P(t) \, \mathrm{d}t = R \, \int_{a}^{b} \left| I(t) \right|^2 \,\mathrm{d}t} .

Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \int_{a}^{b} \left| f(t) \right|^2 \,\mathrm{d}t}

Finite Energy Signals

Complex valued Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} gives amplitude and phase.

Finite energy: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^b \left| f(t) \right|^2 \,\mathrm{d}t < \infty} . Moreover, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\infty}^{\infty} \left| f(t) \right|^2 \,\mathrm{d}t < \infty}

Signals don't need to be continuous (digital circuit signals)

Continuous Signal Space

All signals with finite energy fit into a space:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2 = \left\{ f : \left[ a,b \right] \to \mathbb{C} \mid \int_{a}^{b} \left| f(t) \right|^2 \,\mathrm{d}t < \infty \right\}}

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} comes from mathematician Lebesgue (ca. 1900), and the "square" comes from the square in the integral. The reason for this name is because integral we compute is a Lebesgue integral, not a Riemann integral.

We define an inner product on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle f,g \right\rangle_{L^2[a,b]} = \int_a^b f(t) \, \overline{g(t)} \,\mathrm{d}t}

Discrete Signal Space

Notation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l^2}

By sampling the function at fixed points, we obtain a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \left\{ x_n \right\}_{n=-\infty}^\infty = \left( \dots, x_{-1}, x_0, x_1, \dots \right)}

Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l^2 \subset L^2} .

We define our inner product by a discrete sum:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle X, Y \right\rangle_{l^2} = \sum_{n = -\infty}^\infty x_n \, \bar{y}_n}

The energy in a discrete signal is similarly given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=-\infty}^\infty \left| x_n \right|^2}

Inner Product Spaces

Schwarz's Inequality

Using axioms from definition, we arrive at

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \left\langle \vec{u}, \vec{v} \right\rangle \right| \le \left\| \vec{u} \right\| \, \left\| \vec{u} \right\|} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\| \vec{u} \right\| = \sqrt{\left\langle \vec{u}, \vec{u} \right\rangle}} represents the norm of a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u} \in V} ("LENGTH")

Triangle Inequality

Property of length:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\| \vec{u} + \vec{v} \right\| \le \left\| \vec{u} \right\| + \left\| \vec{v} \right\|} .

"The sum of the lengths any two sides of a triangle must be at least the length of the third side."