Engineering Analysis/Matrices

Norms Induced Norms n-Norm Frobenius Norm Spectral Norm Derivatives Consider the following set of linear equations: We can define the matrix A to represent the coefficients, the vector B as the results, and the vector x as the variables: And rewriting the equation in terms of the matrices, we get: Now, let's say we want the derivative of this equation with respect to the vector x: We know that the first term is constant, so the derivative of the left-hand side of the equation is zero. Analyzing the right side shows us: Pseudo-Inverses There are special matrices known as pseudo-inverses, that satisfies some of the properties of an inverse, but not others. To recap, If we have two square matrices A and B, that are both n × n, then if the following equation is true, we say that A is the inverse of B, and B is the inverse of A: Right Pseudo-Inverse Consider the following matrix: We call this matrix R the right pseudo-inverse of A, because: but We will denote the right pseudo-inverse of A as Left Pseudo-Inverse Consider the following matrix: We call L the left pseudo-inverse of A because but We will denote the left pseudo-inverse of A as