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:

$a_{ij}=0,i\ne j$

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:

$|A-\lambda I|={\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_1{\lambda }^1+a_0$

We can create a companion form matrix in one of two ways:

$\left[\begin{array}{cccccc}0& 0& 0& \cdots & 0& -a_0\\ 1& 0& 0& \cdots & 0& -a_1\\ 0& 1& 0& \cdots & 0& -a_2\\ 0& 0& 1& \cdots & 0& -a_3\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& -a_{n-1}\end{array}\right]$

Or, we can also write it as:

$\left[\begin{array}{cccccc}-a_{n-1}& -a_{n-2}& -a_{n-3}& \cdots & a_1& a_0\\ 0& 0& 0& \cdots & 0& 0\\ 1& 0& 0& \cdots & 0& 0\\ 0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& 0\end{array}\right]$

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:

$\left[\begin{array}{ccccc}a& 1& 0& \cdots & 0\\ 0& a& 1& \cdots & 0\\ 0& 0& a& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& a& \cdots & 1\\ 0& 0& 0& \cdots & a\end{array}\right]$

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:

$\left[\begin{array}{cccc}D& 0& \cdots & 0\\ 0& J_1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & J_n\end{array}\right]$

Or:

$\left[\begin{array}{cccc}{J}_1& 0& \cdots & 0\\ 0& J_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & J_n\end{array}\right]$

The where the D element is a diagonal block matrix, and the J blocks are in Jordan block form.