Eigenvalue Decomposition Eigenvalue decomposition is a fundamental concept in linear algebra that expresses a square matrix as a product of its eigenvectors and a diagonal matrix of its eigenvalues. This decomposition allows a matrix to be diagonalized and simplifies computations involving matrix powers, linear transformations, and analysis of system stability. Definitions Eigenvalue: For a square matrix \( A \), a scalar \( \lambda \) is called an eigenvalue if there exists a nonzero vector \( \vec{v} \) such that: \[ A \vec{v} = \lambda \vec{v} \] Intuitively, this means that multiplying \( A \) by \( \vec{v} \) only scales the vector by \(\lambda\) without changing its direction. Eigenvalues provide important information about the matrix, such as scaling factors in transformations, stability in dynamical systems, and the mod...

Applications of Eigenvalues in Signal Processing Eigenvalues are not just theory — they are central to many core signal processing problems. Below I will explain the main practical applications , with proper mathematics and how they are used in real systems. 1. Principal Component Analysis (PCA) – Signal Compression and Denoising Problem Given noisy signal vectors: \[ \mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_N \] Reduce dimensionality Remove noise Keep maximum signal energy Step 1: Form Covariance Matrix \[ R_x = E[\mathbf{x}\mathbf{x}^T] \] This matrix contains signal correlation information. Step 2: Eigenvalue Decomposition \[ R_x \mathbf{v}_i = \lambda_i \mathbf{v}_i \] \( \lambda_i \) = eigenvalues \( \mathbf{v}_i \) = eigenvectors Interpretation Large eigenvalue indicates direction of high signal energy Small eigenvalue indicates mostly noise Practical Use \[ \mathbf{x}_{approx} = \sum_{i=1}^{k} (\mathbf{v}_i^T \mathb...

Difference Between Eigenvalue, Eigenvector and Eigenfunction 1. Eigenvalue and Eigenvector (Linear Algebra) Definition For a square matrix \( A \): \[ A \mathbf{v} = \lambda \mathbf{v} \] \( \mathbf{v} \neq 0 \) → Eigenvector \( \lambda \) → Eigenvalue \( A \) → Linear transformation The matrix scales the vector without changing its direction. Finding Eigenvalues \[ (A - \lambda I)\mathbf{v} = 0 \] For non-trivial solution: \[ \det(A - \lambda I) = 0 \] This is the characteristic equation . Numerical Example ...

Why a Sine Wave is an Eigenfunction of an LTI System 1. What is an Eigenfunction? A function \( x(t) \) is called an eigenfunction of a system if: \[ \text{System}\{x(t)\} = \lambda x(t) \] Where: The output has the same shape as the input It is only scaled (and possibly phase shifted) \( \lambda \) is the eigenvalue 2. LTI System Description An LTI (Linear Time-Invariant) system is completely described by its impulse response \( h(t) \). The output is given by convolution: \[ y(t) = x(t) * h(t) \] \[ y(t) = \int_{-\infty}^{\...

Why Complex Exponentials Are Eigenfunctions of Every LTI System Eigenfunction of an LTI System For any LTI system , complex exponentials are eigenfunctions. If the input is: $$ x(t) = e^{j\omega_0 t} $$ then the output is: $$ y(t) = H(j\omega_0)\, e^{j\omega_0 t} $$ where \(H(j\omega_0)\) is the system’s frequency response. Why is it called an eigenfunction? Because it satisfies the eigenvalue equation: $$ T\{x(t)\} = \lambda x(t) $$ Eigenfunction → \( e^{j\omega_0 t} \) Eigenvalue → \( H(j\omega_0) \) So the eigenvalue is not \( e^{j\omega_0 t} \). The eigenvalue is the scalar \( H(j\omega_0) \). What about...

To find eigenvalues and eigenvectors: Columns describe how a matrix acts on the coordinate axes, whereas eigenvalues describe how the matrix acts along its own preferred directions (the eigenvectors). Each column is the image of a standard basis vector; only when the standard basis vectors are eigenvectors does the scaling factor equal the corresponding eigenvalue. Columns show what a matrix does to the coordinate axes. Eigenvalues show how the matrix scales its special directions, called eigenvectors. Only when a coordinate axis is an eigenvector does a column get scaled by an eigenvalue. Columns tell us where the x- and y-axes go. Eigenvalues tell us how much the matrix stretches its own special directions (eigenvectors). Eigenvectors and Eigenvalues: Mathematical Example Standard Mathematical Example Given a matrix \( A \) and a vector \( \mathbf{v} \): $$ ...

What is a Hermitian Matrix? A Hermitian matrix is a special type of square matrix that is equal to its own conjugate transpose: \[ A = A^H \] Where \( A^H \) is the conjugate transpose of \( A \) (also known as the adjoint), defined as: \[ A^H = \overline{A}^T \] Here: \( A^T \) is the transpose of \( A \), where rows become columns. \( \overline{A} \) denotes the complex conjugate of the elements of \( A \). For real-valued matrices, the Hermitian property simplifies to the condition that the matrix is equal to its transpose: \[ A = A^T \] Meaning it is a symmetric matrix. Key Properties of Hermitian Matrices The properties of Hermitian matrices are particularly useful in the context of Eigenvalue Decomposition: Real Eigenvalues The eigenvalues of a Hermitian matrix are always real numbers, which is crucial for numerical stab...

Eigenvalue decomposition helps reduce noise by retaining only the dominant (larger) eigenvalues in the eigenvalue matrix while discarding the smaller ones. It decomposes the covariance matrix of the original data using the Principal Component Analysis (PCA) method. The eigenvectors form an orthogonal basis, and the corresponding eigenvalues indicate how much variance (signal) is captured along each eigenvector direction. In signal processing and data science, noise reduction is critical for improving the quality of data. One effective technique for this is Eigenvalue Decomposition (EVD) applied to the covariance matrix of the dataset. Step-by-Step: Noise Reduction with Eigenvalue Decomposition Let’s say you have a dataset represented by a covariance matrix \( C \). Here’s the process mathematically: 1. Perform Eigenvalue Decomposition on \( C \): \[ C = V \Lambda V^T \] Where: \( V \) contains the eigenvectors (p...