Introduction to Matrix Algebra | Civil Crews |
Solving sets of linear simultaneous equations
If A is square then Ax = b has a unique solution
If A is square then Ax = 0 has a non-trivial solution if and only if /A/ = 0
An over-constrained set of equations Ax = b is one in which A has m rows and n columns, where m (the number of equations) is greater than n (the number of variables). The best solution x (in the sense that it minimizes the error /Ax – b/) is the solution of the n equations A’Ax = A’b. If the columns of A are orthonormal vectors then x = A’b.
The Hermitian conjugate of A is A’ = (A*)’, where A* is a matrix each of whose components is the complex conjugate of the corresponding components of A. If A = A’ then A is called a Hermitian matrix.