Engineering Analysis/Matrix Forms

Matrices that follow certain predefined formats are useful in a number of computations. We will discuss some of the common matrix formats here. Later chapters will show how these formats are used in calculations and analysis. Diagonal Matrix A diagonal matrix is a matrix such that: In otherwords, all the elements off the main diagonal are zero, and the diagonal elements may be (but don't need to be) non-zero. Companion Form Matrix If we have the following characteristic polynomial for a matrix: We can create a companion form matrix in one of two ways: Or, we can also write it as: Jordan Canonical Form To discuss the Jordan canonical form, we first need to introduce the idea of the Jordan Block: Jordan Blocks A jordan block is a square matrix such that all the diagonal elements are equal, and all the super-diagonal elements (the elements directly above the diagonal elements) are all 1. To illustrate this, here is an example of an n-dimensional jordan block: Canonical Form A square matrix is in Jordan Canonical form, if it is a diagonal matrix, or if it has one of the following two block-diagonal forms: Or: The where the D element is a diagonal block matrix, and the J blocks are in Jordan block form.