AN INTRODUCTION AND BEGINNER'S GUIDE TO MATRIX PSEUDO-INVERSES,

This paper presents a tutorial development of the theory of matrix pseudo-inverses, with some applications. The proofs are based on two classical theorems - the diagonalization theorem for symmetric matrices and the projection theorem for finite dimensional vector spaces. With the aid of the pseudo-inverse concept, explicit closed form expressions for such things as the general solution to under specified linear equations, the projection of a vector onto a linear manifold, the solution to least squares problems subject to linear constraints and the Gramm-Schmidt orthogon alization procedure are exhibited. An asymptotic expansion of A epsilon B to the minus 1 power, where A is non-negative definite, B is positive definite and epsilon is small the classical perturbation problem which does not require knowledge of the eigenvalues and eigenvectors of A and B is developed. Briefly the definition of the pseudoinverse evolves as follows If H is an n x M matrix and Z is an n-vector, there may or may not be an M-vector X, satisfying the equation HX Z.