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.
A diagonal matrix is a matrix such that:
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{\displaystyle a_{ij}=0,i\neq 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.
If we have the following characteristic polynomial for a matrix:
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{\displaystyle |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:
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{\displaystyle {\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:
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{\displaystyle {\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}}}
To discuss the Jordan canonical form, we first need to introduce the idea of the Jordan Block :
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:
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{\displaystyle {\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}}}
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:
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{\displaystyle {\begin{bmatrix}D&0&\cdots &0\\0&J_{1}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &J_{n}\end{bmatrix}}}
Or:
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{\displaystyle {\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.
