I have a matrix equation and I want to solve it in MATLAB. The equation is
X^(-1)AX=B.
Where X is an unknown symmetric 3x3 matrix, A is known and B is a diagonal matrix (There isn't any condition on B's elements, just diagonal).
Please help me solve this problem!
From the information you've given it appears that you are looking for a symbolic solution which Matlab is not the best at (usually Mathematica is prefered or check out Wolfram Alpha). If you absolutely need Matlab then you could use the 'solve()' command.
However, honestly you are dealing with a rather simple linear system of small dimensions so it is probably easiest to just work this out by hand. Check out Matrix Cookbook for help. Also it really just looks like you are just trying to solve for the eigenvector matrix so possibly the command eig() might help ??
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I'm currently working on solving an equation that involves a symmetric matrix C with 4 unknown variables, and a vector A of the same dimension. The equation I'm trying to solve is:
[C]*{A}=0 (1)
where * denotes matrix multiplication and { } denotes a vector.
The dimension of the C matrix can be as large as 100x100 or more, and I'm trying to define the unknown variables in C in a way that solves equation (1). One approach I've tried is to calculate the determinant of C, |[C]|=0, and solve for the 4 different variables inside.
However, when the dimension of the matrix is large, my current method in Mathematica is not able to solve the problem. I'm wondering if anyone has any suggestions for me to solve this equation more efficiently.
I'm open to using other programming languages such as Matlab or Python as well. Any suggestions or advice would be greatly appreciated.
Thank you in advance for your help and guidance.
I have tried solving this problem with 25*25 dimension. However, this size is not enough for the percision that I want for the variables.
I have a linear equation such as
Ax=b
where A is full rank matrix which its size is 512x512. b is a vector of 512x1. x is unknown vector. I want to find x, hence, I have some options for doing that
1.Using the normal way
inv(A)*b
2.Using SVD ( Singular value decomposition)
[U S V]=svd(A);
x = V*(diag(diag(S).^-1)*(U.'*b))
Both methods give the same result. So, what is benefit of using SVD to solve Ax=b, especially in the case A is a 2D matrix?
Welcome to the world of numerical methods, let me be your guide.
You, as a new person in this world wonders, "Why would I do something this difficult with this SVD stuff instead of the so commonly known inverse?! Im going to try it in Matlab!"
And no answer was found. That is, because you are not looking at the problem itself! The problems arise when you have an ill-conditioned matrix. Then the computing of the inverse is not possible numerically.
example:
A=[1 1 -1;
1 -2 3;
2 -1 2];
try to invert this matrix using inv(A). Youll get infinite.
That is, because the condition number of the matrix is very high (cond(A)).
However, if you try to solve it using SVD method (b=[1;-2;3]) you will get a result. This is still a hot research topic. Solving Ax=b systems with ill condition numbers.
As #Stewie Griffin suggested, the best way to go is mldivide, as it does a couple of things behind it.
(yeah, my example is not very good because the only solution of X is INF, but there is a way better example in this youtube video)
inv(A)*b has several negative sides. The main one is that it explicitly calculates the inverse of A, which is both time demanding, and may result in inaccuracies if values vary by many orders of magnitude.
Although it might be better than inv(A)*b, using svd is not the "correct" approach here. The MATLAB-way to do this is using mldivide, \. Using this, MATLAB chooses the best algorithm to solve the linear system based on its properties (Hermation, upper Hessenberg, real and positive diagonal, symmetric, diagonal, sparse etc.). Often, the solution will be a LU-triangulation with partial permutation, but it varies. You'll have a hard time beating MATLABs implementation of mldivide, but using svd might give you some more insight of the properties of the system if you actually investigates U, S, V. If you don't want to do that, do with mldivide.
Hi I am trying to solve a linear system of the following type:
A*x=b,
where A is the coefficient matrix,
x is the vectors of unknowns and
b is the vector of solution.
The coefficient matrix (A) is a n-by-n sparse matrix, with even zeros in the diagonal. In order to solve this system in an accurate way I am using an iterative method in Matlab called bicgstab (Biconjugate gradients stabilized method).
This coefficient matrix (A) has a
det(A)=-4.1548e-05 and a rcond(A)= 1.1331e-04.
Therefore the matrix is ill-conditioned. I first try to perform a scaling and the results where:
det(A)= -1.2612e+135 but the rcond(A)=5.0808e-07...
Therefore the matrix is still ill-conditioned... I verify and the sum of all absolute value of the non-diagonal elements where 163.60 and the sum of all absolute value of the diagonal elements where 32.49... Therefore the matrix of coefficient is not diagonally dominant and will not converge using my function bicgstab...
I am looking for someone that can help me with performing a pivoting to the coefficient matrix (A) so it can be diagonally dominant. Or any advice to solve this problem....
Thanks for the help.
First there should be quite a few things noted here:
Don't use the determinant to estimate the "amount of singularity" of your matrix. The determinant is the product of all the eigenvalues of your matrix, and therefore its scaling can be wildly misleading compared to a much better measure like the condition number, leading to the next point..
your conditioning (according to rcond) isn't that bad, are you working with single or double precision? Large problems can routinely get condition numbers in this range and still be quite solvable, but of course this depends on a very complicated interaction of many factors, of which the condition number plays only a small part. This leads to another complicated point:
Diagonal dominance may not help you at all here. BiCGStab as far as I know does not require diagonal dominance for its convergence, and also I don't think diagonal dominance is known even to help it. Diagonal dominance is usually an assumption made by other iterative methods such as the Jacobi method or Gauss-Seidel. Actually the convergence behavior of BiCGStab is not very well understood at all, and it is usually only used when memory is a very severe problem but conjugate gradients is not applicable.
If you are really interested in using a Krylov method (such as BiCGStab) to solve your problem, then you generally need to have more understanding of where your matrix is coming from so that you can choose a sensible preconditioner.
So this calls for a bit more information. Do you know more about this matrix? Is it arising from some kind of physical problem? Do you know for example if it is symmetric or positive definite (I will assume not both because you are not using CG).
Let me end with some actionable advice which is very generic, and so not necessarily optimal:
If memory is not an issue, consider using restarted GMRES instead of BiCGStab. My experience is that GMRES has much more robust convergence.
Try an approximate factorization preconditioner such as ILU. MATLAB has a function for this built in.
I have a linear system of equations AX = B to solve in MATLAB. What I have known is A is sparse, positive-definite and symmetric. I know the command x = A \ b works yet I am not sure MATLAB takes full advantage of A's good properties so as to maximize the efficiency. Is there any way to specify the algorithm to solve it, for example Conjugate Gradient algorithm in MATLAB?
If your matrix is sparse, you can use all these iterative functions, for example bicg for a biconjugate gradients method.
MATLAB's mldivide operator does indeed take advantage of properties of A. See the documentation for details - expand the "Algorithm" section.
I'm trying to find eigenvalues of a matrix without using eig function (my homework says so). In Matlab, I define the matrix and identity matrix. But I cannot set up this equation:
A - x*I
x here is lambda, A is the matrix that I should find eigenvalues of and I is the identity matrix. If you know how to find eigenvalues, you supposed to understand this. How can I go through?
you can get some inspiration here: http://en.wikipedia.org/wiki/Eigenvalue_algorithm
if the matrix is fixed size, you can easily do the det(A-lambda*eye)=0 solving by yourself and use that.
With power iteration you can already find the dominant eigenvalue, and I knew there was an extension to this algorithm to also find the other eigenvalues, but cannot recall how that works :(