Spectral Theorem Applications & Proof

Applications of Spectral Theorem

• Orthonormal basis exists for a normal operator $T$ on inner product space $V$, consisting of $T$'s eigenvectors.
• For distinct eigenvalues $\lambda_1,..., \lambda_k$ of $T$, there exist orthogonal projections $E_1,..., E_k$ s.t. $T = \sum_{i=1}^{k} \lambda_i E_i$.
• $E_i$ is the orthogonal projection onto $W_i$, the eigenspace of $\lambda_i$.
• $E_i E_j = 0$ if $i \neq j$
• $\sum_{i=1}^{k} E_i = I$ (Identity operator)
• $(\lambda_i, E_i)$ are unique.

Corollary

• If $T$ is a normal operator, $T^* = g(T)$ for some polynomial $g$.

Proof of Corollary

• $T = \sum_{i=1}^{k} \lambda_i E_i$
• $T^* = \sum_{i=1}^{k} \overline{\lambda_i} E_i$
• A polynomial $g$ needs to be found such that $g(\lambda_i) = \overline{\lambda_i} ; \forall i$
• A system of equations can be formed: $a_0 + a_1 \lambda_i +... + a_{k-1} \lambda_i^{k-1} = \overline{\lambda_i}$ for $i = 1,..., k$
• $g(z) = a_0 + a_1 z +... + a_{k-1} z^{k-1}$ represents the polynomial
• The matrix equation is: $V \mathbf{a} = \overline{\mathbf{\lambda}}$, where $V$ is a Vandermonde matrix.
• $det(V) = \prod_{i < j} (\lambda_j - \lambda_i) \neq 0$, given that $\lambda_i$ are distinct
• A unique solution $(a_0,..., a_{k-1})$ exists.
• $g(T) = a_0 I + a_1 T +... + a_{k-1} T^{k-1} = \sum_{i=1}^{k} g(\lambda_i) E_i = \sum_{i=1}^{k} \overline{\lambda_i} E_i = T^*$

Theorem

• For a normal operator $T$ on $V$, there exists a normal operator $N$ such that $N^2 = T$.

Proof

• $T = \sum_{i=1}^{k} \lambda_i E_i$
• Find $\mu_i \in \mathbb{C}$ such that $\mu_i^2 = \lambda_i$.
• $N = \sum_{i=1}^{k} \mu_i E_i$
• $N^2 = (\sum_{i=1}^{k} \mu_i E_i) (\sum_{j=1}^{k} \mu_j E_j) = \sum_{i,j} \mu_i \mu_j E_i E_j = \sum_{i=1}^{k} \mu_i^2 E_i = \sum_{i=1}^{k} \lambda_i E_i = T$
• $N^* = \sum_{i=1}^{k} \overline{\mu_i} E_i$
• $NN^* = (\sum_{i=1}^{k} \mu_i E_i) (\sum_{j=1}^{k} \overline{\mu_j} E_j) = \sum_{i=1}^{k} |\mu_i|^2 E_i$
• $N^*N = (\sum_{i=1}^{k} \overline{\mu_i} E_i) (\sum_{j=1}^{k} \mu_j E_j) = \sum_{i=1}^{k} |\mu_i|^2 E_i$
• Therefore, $NN^* = N^*N$, proving $N$ is normal.

Polar Decomposition

• Given a linear operator $T$ on $V$, the following are equivalent:
  • There exists an isometry $S$ and positive operator $P$ such that $T = SP$.
  • There exists an isometry $S'$ and positive operator $P'$ such that $T = P'S'$.
  • $T$ is invertible.

Proof

• $(a) \implies (c)$:
  • $T = SP$
  • $P$ is positive, so $P$ is self-adjoint and all eigenvalues $\lambda_i > 0$, implying $P$ is invertible.
  • $S$ is an isometry, so $S$ is invertible.
  • $T$ is invertible since the product of invertible maps is invertible.
• $(c) \implies (a)$:
  • $T^*T$ is a positive operator because $\langle T^*T v, v \rangle = \langle T v, T v \rangle \geq 0$.
  • $T^*T$ is invertible because the product of invertible maps is invertible.
  • All eigenvalues of $T^*T$ are $> 0$.
  • Let $P = \sqrt{T^*T}$ (exists by theorem), $P$ is positive.
  • Defining $S = TP^{-1}$, then $T = SP$.
  • $S$ is shown to be an isometry through inner product preservation: $\langle Sv, Sw \rangle = \langle TP^{-1}v, TP^{-1}w \rangle ... = \langle v, w \rangle$.
• $(c) \implies (b)$:
  • Similarly, $TT^*$ is positive and invertible, so all eigenvalues are $> 0$.
  • Let $P' = \sqrt{TT^*}$ (exists by theorem), giving that $P'$ is positive.
  • Define $S' = (P')^{-1}T$, then $T = P'S'$.
  • $S'$ is an isometry is shown via inner product: $\langle S'v, S'w \rangle = \langle (P')^{-1}T v, (P')^{-1}T w \rangle = ... = \langle T^Tv, (TT^)^{-1} T w \rangle $

Time-Frequency Analysis: Why?

• Fourier transform indicates frequencies, but not their timing
• Time-frequency analysis shows frequency changes over time

Time-Frequency Analysis: How?

• Uses Short-Time Fourier Transform (STFT)
• Uses Wavelet Transform

Short-Time Fourier Transform (STFT)

• Divides signal into short segments
• Computes Fourier transform of each segment
• Formula: $X(t, f) = \int_{-\infty}^{\infty} w(t - \tau)x(\tau)e^{-j2\pi f\tau}d\tau$
  • $x(\tau)$: signal
  • $w(t - \tau)$: window function at time $t$
  • $X(t, f)$: STFT of signal

Windowing in STFT

• Reduces spectral leakage
• Improves STFT accuracy
• Spectral leakage: energy spreads into other frequencies
• Window function: weights the signal prior to Fourier transform
  • Rectangular window: simplest, poor frequency resolution
  • Hamming window: good general-purpose, good frequency resolution and low spectral leakage
  • Gaussian window: best time-frequency resolution, poor frequency resolution

Spectrogram Definition

• Spectrogram represents magnitude squared of STFT: $S(t, f) = |X(t, f)|^2$
• Illustrates signal power at time and frequency points

Time-Frequency Resolution: STFT

• Determined by window function width
  • Wide window: good frequency resolution, poor time resolution
  • Narrow window: good time resolution, poor frequency resolution
• Uncertainty principle: $\Delta t \Delta f \geq \frac{1}{4\pi}$
  • Defines trade-off between resolutions

Wavelet Transform

• Time-frequency analysis using wavelet function
• Decomposes signal into frequency components
• $\gamma(t, f) = \int_{-\infty}^{\infty} \psi_{t,s}^{*}(\tau)x(\tau)d\tau$
  • $x(\tau)$: signal
  • $\psi_{t,s}(\tau) = \frac{1}{\sqrt{s}}\psi(\frac{\tau - t}{s})$: wavelet function
  • $t$: translation parameter (time location)
  • $s$: scale parameter (width)
  • $\gamma(t, f)$: wavelet transform

Wavelet Function Details

• Localized in time and frequency
  • Examples: Haar, Daubechies, Morlet wavelets
  • Properties:
    • Zero mean: $\int_{-\infty}^{\infty} \psi(t) dt = 0$
    • Normalized: $\int_{-\infty}^{\infty} |\psi(t)|^2 dt = 1$

Time-Frequency Resolution: Wavelet Transform

• Controlled by scale parameter
  • Small scale: good time resolution, poor frequency resolution
  • Large scale: good frequency resolution, poor time resolution

STFT vs Wavelet Transform

• STFT has fixed window, Wavelet Transform has variable window
• STFT has fixed resolution, Wavelet Transform has variable resolution
• Wavelet Transform is better for non-stationary signals

Applications of Time-Frequency Analysis

• Useful in speech and music analysis
• Also, useful in medical and geophysical signal processing and image processing

Vector Fields in the Plane

• Definition: Function assigning 2D vectors to points in $\mathbb{R}^2$
• Notation: $\mathbf{F}(x, y) = \langle P(x, y), Q(x, y) \rangle$

Examples of Vector Fields

• Fluid flow: $\mathbf{V}(x, y)$ represents fluid velocity.
• Gravitational field: $\mathbf{F}(x, y)$ represents gravitational force.
• Electric field: $\mathbf{E}(x, y)$ represents electric force.

Sketching Vector Fields

• Plot vectors at points $(x, y)$
• Example: $\mathbf{F}(x, y) = \langle y, x \rangle$

Vector Fields in Space

• Definition: Function assigning 3D vectors to points in $\mathbb{R}^3$
• Notation: $\mathbf{F}(x, y, z) = \langle P(x, y, z), Q(x, y, z), R(x, y, z) \rangle$

Examples of Vector Fields in Space

• Gravitational field: $\mathbf{F}(x, y, z)$
• Electromagnetic field: $\mathbf{E}(x, y, z)$

Gradient Vector Fields

• Gradient of scalar function $f(x, y)$: $\nabla f(x, y) = \langle f_x(x, y), f_y(x, y) \rangle$
• Gradient of scalar function $f(x, y, z)$: $\nabla f(x, y, z) = \langle f_x(x, y, z), f_y(x, y, z), f_z(x, y, z) \rangle$

Conservative Vector Fields

• Vector field $\mathbf{F}$ is conservative if $\mathbf{F} = \nabla f$ for some scalar function $f$.
• $f$ is the potential function.

Example: Finding Potential Function

• Given $\mathbf{F}(x, y) = \langle 2x + y^2, 2xy \rangle$, find $f(x, y)$.
• Solution:
  • $f_x(x, y) = 2x + y^2$, integrating gives $f(x, y) = x^2 + xy^2 + g(y)$.
  • $f_y(x, y) = 2xy + g'(y)$, comparing with given $f_y(x, y) = 2xy$ gives $g'(y) = 0$.
  • $g(y) = K$, thus $f(x, y) = x^2 + xy^2 + K$ is the potential function.

Reaction Rate

• Is the speed at which reactants convert to products

Factors Affecting Reaction Rate

• Concentration of reactants: Higher concentration increases the reaction rate.
• Temperature: Higher temperature generally increases reaction rate.
• Catalysts: These speed up reactions without being consumed
• Surface Area: Increased surface area increases reaction rate for solids.

Rate Law

• Equation expressing reaction rate using reactant concentrations

General Form of Rate Law

• Given $aA + bB \rightarrow cC + dD$, $rate = k[A]^m[B]^n$:
  • $k$ is the rate constant
  • $[A]$ and $[B]$ are the concentrations of reactants A and B
  • $m$ and $n$ are the reaction orders with respect to A and B, respectively
  • $m + n$ is the overall reaction order

Determining Reaction Order

• Reaction orders ($m$ and $n$) are determined experimentally

Integrated Rate Laws

• Zero-Order Reactions
  • $rate = k$
  • $[A] = -kt + [A]_0$
  • $t_{1/2} = \frac{[A]_0}{2k}$
• First-Order Reactions
  • $rate = k[A]$
  • $ln[A] = -kt + ln[A]_0$
  • $t_{1/2} = \frac{0.693}{k}$
• Second-Order Reactions
  • $rate = k[A]^2$
  • $\frac{1}{[A]} = kt + \frac{1}{[A]_0}$
  • $t_{1/2} = \frac{1}{k[A]_0}$
• $[A]$ is the concentration of reactant A at time $t$
• $[A]_0$ is the initial concentration of reactant A
• $k$ is the rate constant
• $t_{1/2}$ is the half-life

Activation Energy ($E_a$)

• Minimum energy required for a chemical reaction to occur

Arrhenius Equation

• $k = Ae^{-E_a/RT}$ relates rate constant ($k$) to $E_a$ and temperature ($T$):
  • $A$ is the pre-exponential factor
  • $R$ is the ideal gas constant $(8.314 J/(mol \cdot K))$
  • $T$ is the absolute temperature (in Kelvin)

Determining Activation Energy

• $lnk = lnA - \frac{E_a}{RT}$ yields a straight line with plotting $lnk$ versus $\frac{1}{T}$

Reaction Mechanisms

• Step-by-step sequence of elementary reactions by which overall chemical change occurs

Elementary Reactions

• Single-step reactions that cannot be broken down into simpler steps

Rate-Determining Step

• Slowest step in a reaction mechanism, determines overall reaction rate

Intermediates

• Species produced and consumed during reaction mechanism that is not present in the overall balanced equation

Catalysis

• Speeding up a chemical reaction by adding a catalyst

Types of Catalysis

• Homogeneous: Catalyst and reactants are the same phase
• Heterogeneous: Catalyst and reactants are different phases

How Catalysts Work

• Lower the activation energy of a reaction
• Provide an alternative reaction pathway