How to convert polynomial to matrix?

[m2006]

How to convert polynomial to matrix?

Does anyone know how to convert a polynomial to a matrix. I have been told that it is possible to convert a fourth order polynomial to a 4x4 Matrix in Matlab but I have no clue how to go about it and how it is done. Any suggestions or even where to look would be GREATLY helpful!

 

[billk]

Can you be more specific about what you are trying to do?

If you were to choose a basis for the space of fourth degree polynomials, like $$\{ 1,x,x^2,x^3,x^4 \}$$, then it is possible to represent a polynomial as an element of this vector space by a coefficient matrix.

For example, $$3x^4-2x^2+5 \mapsto \left[5,0,-2,0,3 \right]$$. This is not a 4x4 matrix like you are looking for, but it is a representation of the polynomial as a matrix.

If you are more specific about what you are trying to do, maybe I can offer more insight, although I am not sure if what you want can be done.

 

[thamster]

check this out for making a companion matrix for a given polynomial:
http://en.wikipedia.org/wiki/Companion_matrix

The determinant of a companion matrix is a polynomial in λ, known as the characteristic polynomial. λs are the eigenvalues, they are also the solutions to the polynomial.

This works well for polynomials of degree 4 or smaller since they can be solved by a finite sequence of arithmetic operations and radicals.

hope this helps.

 

[spg89]

use cayley-Hamilton theorem from backwards,it will help...

 

[blue_raver22]

There is a method called "companion matrix" that can be used to convert a polynomial to a matrix. This method involves creating a matrix with specific patterns of coefficients that correspond to the coefficients of the polynomial. The resulting matrix will have a characteristic polynomial that is equivalent to the original polynomial.

To convert a fourth-order polynomial to a 4x4 matrix in Matlab, you can use the command "compan(p)" where "p" is the vector of coefficients of the polynomial in descending order. This will create a companion matrix that you can then manipulate and use in your calculations.

You can also refer to online resources or textbooks for more information and examples on how to convert polynomials to matrices using the companion matrix method.