Engineering Analysis/Matrix Forms

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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[edit]

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[edit]

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:

\begin{bmatrix} 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 \\
                       \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\
                       0 & 0 & 0 & \cdots & 1 & -a_{n-1} 
       \end{bmatrix}

Or, we can also write it as:

\begin{bmatrix} -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 \\
                       \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\
                       0 & 0 & 0 & \cdots & 1 & 0 
       \end{bmatrix}

Jordan Canonical Form[edit]

To discuss the Jordan canonical form, we first need to introduce the idea of the Jordan Block:

Jordan Blocks[edit]

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:

\begin{bmatrix} a & 1 & 0 & \cdots & 0 \\
                       0 & a & 1 & \cdots & 0 \\
                       0 & 0 & a & \cdots & 0 \\
                       \vdots & \vdots & \vdots &\ddots & \vdots \\
                       0 & 0 & a & \cdots & 1 \\
                       0 & 0 & 0 & \cdots & a 
       \end{bmatrix}

Canonical Form[edit]

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:

\begin{bmatrix}D & 0 & \cdots & 0 \\
                      0 & J_1 & \cdots & 0 \\
                      \vdots & \vdots &\ddots & \vdots \\
                      0 & 0 & \cdots & J_n
       \end{bmatrix}

Or:

\begin{bmatrix}J_1 & 0 & \cdots & 0 \\
                      0 & J_2 & \cdots & 0 \\
                      \vdots & \vdots &\ddots & \vdots \\
                      0 & 0 & \cdots & J_n
       \end{bmatrix}

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