Matrix equation - numeric solving

In summary, the conversation discusses a problem involving the numerical counting of "lambda" in an equation with matrices and an unknown vector. The equation can be turned into a first order eigenproblem and solved using symbolic calculations or a library like GSL. The discussion also mentions a possible solution involving matrices with nicer properties.
  • #1
Anthus
14
0
Hello,
my problem is to numerically count "lambda" from that kind of equation:
[tex](\lambda ^2 \check{A} + \lambda \check{B} + \check{C}) \vec{v}=0
[/tex]
which means that:
[tex]det(\lambda ^2 \check{A} + \lambda \check{B} + \check{C})=0
[/tex]
where A, B and C are n x n matrices, v is unknown vector and "lambda" is unknown.

It is easy to solve it in Matlab using symbolic calculations of determinant to get polynomial, but I write my own program so I can't use it. I have i.e. GSL library. This equation without first or second term is generalized eigensystem, but perhaps not real.
Does anybody know what use to solve that?
 
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  • #2
You can turn this into a "first order" eigenproblem with the matrices twice the size

[tex]\lambda\begin{pmatrix}A & 0 \cr 0 & I \end{pmatrix}
\begin{pmatrix}\lambda v \cr v \end{pmatrix}
+ \begin{pmatrix}B & C \cr -I & 0\end{pmatrix}
\begin{pmatrix}\lambda v \cr v \end{pmatrix} = 0[/tex]

This works nicely when A is symmetric and/or positive definite.

If matrix C has "nicer" properties than A, you can make a similar eigenproblem with eigenvalues [itex]1/\lambda[/itex]

Note, the [itex]\lambda[/itex] in the eigenvector is not an issue. So far as the numerical solution of the eigenproblem is concerned, the eigenvector is just an arbitrary vector with 2n elements, and the fact that the top half of the vector happens to be a scalar multiple of the bottom half is just a conincidence.
 
  • #3
Thank you, I'll think about it :)
 

1. What is a matrix equation?

A matrix equation is a mathematical representation of a system of linear equations using matrices. It is written in the form Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix.

2. How is a matrix equation solved numerically?

A matrix equation is solved numerically by using various methods such as Gaussian elimination, LU decomposition, and iterative methods like Jacobi or Gauss-Seidel. These methods involve manipulating the matrices to reduce the equation to a simpler form until a solution can be found.

3. What is the importance of solving matrix equations?

Matrix equations are used in various fields of science and engineering to model and solve complex systems. They are important in numerical analysis, statistics, physics, and many other areas. Solving matrix equations allows us to find solutions to real-world problems and make predictions based on mathematical models.

4. What are some common applications of matrix equations?

Matrix equations are used in image processing, optimization problems, computer graphics, and many other areas of science and technology. They are also used in solving systems of differential equations, which are commonly used to model physical systems in physics and engineering.

5. Can a matrix equation have more than one solution?

Yes, a matrix equation can have one, infinite, or no solutions. It depends on the properties of the coefficient matrix and the constants in the equation. A system of linear equations has one unique solution if the determinant of the coefficient matrix is non-zero. If the determinant is zero, the system may have no solution or an infinite number of solutions.

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