Solve Weird Matrix Equation to Get Values for P

In summary, the conversation revolved around solving systems of ODE using Laplace transforms and the discovery of a matrix that could transform the DE into another form. The possibility of finding a unique P value was discussed, but it was determined that such a value does not exist due to the different eigenvalues of the left and right numerical matrices. The conversation then turned to the possibility of using a reflection matrix to solve the problem and the general approach for finding a unique P value in the case of distinct eigenvalues. Ultimately, it was concluded that the reflection matrix J could be used to solve the problem.
  • #1
maistral
240
17
So while I was trying to browse some notes with regard to Laplace transforms in solving systems of ODE, this matrix came up:
matrix eqn.png

I could easily just use RK4 and call it a day to nuke the systems of ODE, but this actually made me curious. Apparently one can transform the matrix DE into another form, then this appeared.

How do I get the values for P?
 
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  • #2
I don't think such ##P## exists. The numerical matrices on the left and right have different eigenvalues.
 
  • #3
S.G. Janssens said:
I don't think such ##P## exists. The numerical matrices on the left and right have different eigenvalues.
It exists and is not unique.

I was able to obtain a possible ##P## using Mathematica. However, I do not know how to do it in a rigorous fashion.
 
  • #4
What do you find for the eigenvalues of the left and right numerical matrices? For the one on the left I get one real eigenvalue and a complex-conjugate pair, while for the matrix on the right I get ##\{-3,-2,-1\}##. So these two matrices cannot be similar.

(If such ##P## would exist, it would not be unique indeed, because every nonzero multiple of ##P## would also qualify.)
 
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  • #5
S.G. Janssens said:
What do you find for the eigenvalues of the left and right numerical matrices? For the one on the left I get one real eigenvalue and a complex-conjugate pair, while for the matrix on the right I get ##\{-3,-2,-1\}##. So these two matrices cannot be similar.
I made a mistake and used +11 instead of -11 in the right matrix. My guess is that there is an error in the OP and that the two matrices differ only by the arrangement of the numerical values.
 
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  • #6
assuming the stated typo, consider using the reflection matrix

##\mathbf J = \left[\begin{matrix}0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{matrix}\right]##

which is a permutation matrix (and involutary)

note:
For the nice case of diagonalizable matrices with the same spectrum (and in particular the extra nice case where all eigenvalues are distinct), the general approach is
##\mathbf S^{-1} \mathbf A \mathbf S= \mathbf D ##
and
##\mathbf U^{-1} \mathbf B \mathbf U= \mathbf D ##

so
##\mathbf U^{-1} \mathbf B \mathbf U= \mathbf S^{-1} \mathbf A \mathbf S ##
## \mathbf B = \mathbf U\mathbf S^{-1} \mathbf A \mathbf S\mathbf U^{-1} ##
setting ##\mathbf P:= \mathbf S\mathbf U^{-1}##
gives
## \mathbf B = \mathbf P^{-1} \mathbf A \mathbf P ##

the fact that the eigenvalues are distinct proves that ##\mathbf P## is unique (up to rescaling)

but in this case I could just eyeball the problem as a graph isomorphism and come up with ##\mathbf J##
 
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Related to Solve Weird Matrix Equation to Get Values for P

1. What is a matrix equation?

A matrix equation is a mathematical expression that involves matrices, which are rectangular arrays of numbers or variables. The equation typically involves multiplying one or more matrices together to get a final result.

2. What is a "weird" matrix equation?

A "weird" matrix equation is one that may be difficult to solve or involves non-standard operations, such as taking the inverse or determinant of a matrix. It may also involve complex numbers or variables.

3. Why is it important to solve a matrix equation?

Solving a matrix equation can help us find the values of variables or parameters in a system of equations. This is useful in many fields, including physics, engineering, and economics, where systems of equations are commonly used to model real-world situations.

4. How do I solve a matrix equation to get values for P?

The process for solving a matrix equation to get values for P will depend on the specific equation. In general, you will need to use algebraic manipulation and matrix operations, such as multiplying, adding, and subtracting matrices, to isolate the variable P on one side of the equation. You may also need to use techniques like Gaussian elimination or matrix inversion to simplify the equation and solve for P.

5. What are some tips for solving a weird matrix equation?

Some tips for solving a weird matrix equation include: breaking the problem down into smaller steps, using properties of matrices to simplify the equation, checking your work for errors, and seeking help from a tutor or online resources if you get stuck. It can also be helpful to have a good understanding of basic matrix operations and algebraic principles.

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