Applications of Linear Algebra
Introduction
We will provide a practical review of linear algebra concepts and usage while skipping some introduction to linear algebra operations.
The following topics are discussed in this article: inner product, norm, eigenvalue decomposition, diagonalization of matrices, matrix norm
Inner Product
Definition: An inner product on a vector space is a function that takes two vectors and returns a scalar, typically denoted as ⟨x,y⟩. It generalizes the dot product in Euclidean space.
Examples and Applications:
Geometry:
In Euclidean space, the inner product simplifies to the dot product, expressed as x⋅v=||x||*||v||cos(α), where α is the angle between vectors x and v.
Probability and Statistics:
The expectation of a random variable X, denoted E[X], can be interpreted as an inner product between the range of X and its associated probability density function PDF. The result of this expectation is a scalar value which in this case is the average value that one expects this random variable X will take.
Signal Processing - Convolution Operation:
The convolution of two functions H(t) and X(t) is analogous to a sliding window inner product between H(−t+n) and X(t), where n represents the shifting term. Note the negative sign in H(−t); it indicates a flipped version of the window function h(t). In signal processing h(t) typically represents the impulse response of a linear system.
Norm
Definition: The norm of a vector x, denoted as ∥x∥, is a measure of the 'length' of the vector. In an inner product space, it is defined as
∥x∥= ⟨x, x⟩
Eigenvalue Decomposition
Concept: If A is a square matrix, an eigenvalue decomposition finds matrices P and D such that A=PDP ^−1 where D is diagonal and P contains the eigenvectors of A. Eigenvalues and Eigenvectors: An eigenvalue λ and eigenvector v of A satisfy Av=λv.
Applications: Used in solving linear differential equations, PCA in machine learning, and more.
Diagonalization of Matrices
Definition: A matrix A is diagonalizable if it can be transformed into a diagonal matrix through a similarity transformation (i.e., preservation of eigenvalues before and after the transformation). This means there exists a matrix P such that performing the operation P^−1AP results in a diagonal matrix, which we will call E. In E, only the diagonal elements contain values, and all other elements are zeros.
Note: In linear algebra, a transformation typically refers to changing the coordinate system through the multiplication by a transformation matrix. This process alters the representation of vectors or matrices without changing their intrinsic properties. Transformation Matrix: The matrix P is used to perform this transformation. It acts as a tool to shift from one coordinate system to another.
A specific type of transformation is similarity transformation, which is a process where a matrix A is converted into another matrix B using a particular form of matrix multiplication, involving a matrix P and its inverse. This process is described by the formula B=P^−1AP.
Condition: A is diagonalizable if it has n linearly independent eigenvectors, where n is the dimension of the matrix
Implication and Applications
Simplification of Computations:
Diagonal matrices greatly simplify matrix computations. For example, consider a diagonal matrix E as a scaled version of an identity matrix. When multiplying another matrix Q by E, each row of Q gets scaled by the corresponding diagonal element of E.
Similarly, when multiplying a vector V by a diagonal matrix E, the operation can be visualized as an element-wise multiplication of the vector elements with the corresponding diagonal elements of E
Connection to Frequency Domain Filtering:
This concept resembles filtering in the frequency domain. In a special case, where the matrix P is the Discrete Fourier Transform (DFT) matrix and A is a circulant matrix constructed from impulse response coefficients, the diagonal elements of the resultant matrix E are the DFT of the impulse response. This relationship is a cornerstone in signal processing, particularly in the context of linear time-invariant systems.
Example
Circulant Matrix: A circulant matrix is an example of a matrix that can be diagonalized using the DFT matrix. This property is extensively used in signal processing for efficient computation of linear convolutions and correlations.
Matrix Norm
Definition: A matrix norm is a natural extension of vector norms to matrices, providing a measure of the 'size' of the matrix.
Types:
Frobneious Norm: it is the square root of the sum of the absolute squares of its elements of matrix A
Spectral Norm (2-norm): it is the largest singular value of matrix A
Nuclear Norm (trace-norm): It is the sum of its singular values of matrix A