Practice Math With Us: Polynomial Matrices

Tuesday, May 25, 2010

Polynomial Matrices

In this lesson let try to understand more on A polynomial matrices or matrix polynomial is a matrix whose elements are univariate or multivariate polynomials.

Properties of Polynomial Matrices
Following are the properties of polynomial Matrics and its functions

A polynomial matrix in excess of a field with determinant equivalent to a non-zero constant is called uni modular, and have an inverse, which is also a polynomial matrix.
Note, that the simply scalar uni modular polynomials are polynomials of degree 0 - nonzero constants, for the reason that an inverse of an arbitrary polynomial of high degree is a rational function.
The roots of a polynomial matrix in excess of the complex numbers are the points in the complex plane wherever the matrix loses rank.

Example 1
An example the 3x3 polynomial matrices
$\displaystyle{P}={\left[\matrix{{1}&{{x}}^{{{2}}}&{x}\\{0}&{2}{x}&{2}\\{8}{x}+{2}&{{x}}^{{2}}-{1}&{0}}\right]}$

$\displaystyle={\left[\matrix{{1}&{0}&{0}\\{0}&{0}&{2}\\{2}&-{1}&{0}}\right]}$ + $\displaystyle{\left[\matrix{{0}&{0}&{1}\\{0}&{2}&{0}\\{3}&{0}&{0}}\right]}{x}+$ $\displaystyle{\left[\matrix{{0}&{1}&{0}\\{0}&{0}&{0}\\{0}&{1}&{0}}\right]}{\times}^{{{2}}}$
This is how we solve problems on plynomial Matrices. Hope this lesson is more useful to you. Keep reading.... and leave your comments if you wish to share anything to me . May be in the following lesson we shall dicuss on Various topics involved in the algebra ii

Practice Math With Us

Tuesday, May 25, 2010

Polynomial Matrices

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