How can i split a matrix into product of two matrices in matlab?
21 visualizaciones (últimos 30 días)
Mostrar comentarios más antiguos
Betha Shirisha
el 15 de Feb. de 2015
Comentada: John D'Errico
el 15 de Feb. de 2015
I am having a matrix A which is formed by multplying matrix X with it's conjugate transpose (A=X*X'),how can i find matrix X from A?
0 comentarios
Respuesta aceptada
emehmetcik
el 15 de Feb. de 2015
@John D'Errico
You are right about your comment on singularity of a matrix and positive definiteness.
I assumed that since it was a covariance matrix, it should be positive semi definite. Since the error message is `Matrix must be positive definite`, I concluded that the determinant of the matrix A should be equal to zero and therefore it is not positive definite (it is positive semi definite). So the conclusion must have been that the matrix A is singular, rather than it is not positive definite (although it is still a correct proposition).
What I should have said was:
If the A matrix is not positive definite, ...
"it means that the matrix A is singular, (because it is a covariance matrix, therefore it is positive semi definite)"
... which means that the solution you are looking for is not unique. That is to say, there are many X matrices satisfying A = X*X'.
Hence I was talking about this specific problem, rather than trying to point out a general fact. Sorry for the confusion that I created.
...
But I think the error here is not caused by a complex diagonal element, as you mentioned above, since A = X * X' and as I said many times already it is a covariance matrix. And the error message returned by chol can also be generated by real diagonal inputs as well, by using a singular matrix;
chol([1, 2; 3, 4])
Error using chol
Matrix must be positive definite.
Therefore, my suggestion to use svd will not fail for this problem, because we are dealing with a singular "A" matrix, which is a covariance matrix (A = X*X'), therefore it is Hermitian and is diagonalizable.
But in general, I agree that not every matrix can be decomposed into the form described above (A = X*X'), as your example of a matrix with complex diagonal entry suggests.
4 comentarios
John D'Errico
el 15 de Feb. de 2015
I will only repeat the statement that the error:
[R] = chol(WW)
Error using chol Matrix must be positive definite with real diagonal.
is generated ONLY when a complex diagonal element is present, and under apparently no other conditions that I can find. (Since chol is built-in, we cannot see the code.) If the matrix is merely singular, even if the off-diagonal elements are complex, you will see a different error.
Whether the matrix MAY be numerically singular in addition to that fact, I cannot know myself, since I have not seen the matrix. The last comment by Betha indicates the 4x4 matrix probably has numerical rank 2 since it has two tiny singular values.
And, no, there is no unique solution when you have a singular problem.
Más respuestas (1)
John D'Errico
el 15 de Feb. de 2015
help chol
5 comentarios
John D'Errico
el 15 de Feb. de 2015
Editada: John D'Errico
el 15 de Feb. de 2015
@emehmetcik - NO. Your statement is simply not correct. You are possibly confusing the idea of positive definite with singularity for a matrix.
A singular matrix will have infinitely many solutions.
A positive definite matrix M has the property that for any possible vector x, the product (x'*M*x) will be a positive number.
This is sometimes confused by people, because if M is a singular matrix, then there exists some vector x such that M*x will yield the zero vector.
The error message returned by chol is a reflection of the requirement for chol, that it requires a matrix with real diagonal.
Your suggestion to use SVD will fail. Why? Because that error message is returned when the matrix has a complex diagonal element.
As you can see, the error message that is generated indicates a complex diagonal element was supplied.
chol(diag([1 0]))
Error using chol
Matrix must be positive definite.
chol(diag([i 0]))
Error using chol
Matrix must be positive definite with real diagonal.
Only when a diagonal element was not real did chol produce that message.
So the question is, can you find a factorization using the scheme you propose using SVD (or any scheme) for a matrix with diagonal elements that are not real? The simple answer is no, and in fact, that is easily proven.
Consider what the diagonal elements of a matrix as a product X*X' are formed from. The diagonal element (so the (k,k) element) of the matrix:
M = X*X'
will be
X(k,:)*X(k,:)'
So the dot product of a vector with itself, conjugate transposed. Consider that this result will NEVER be a complex number.
Therefore SVD will NEVER yield a solution to a problem that has no solution possible.
In addition, your suggestion to use cholcov will also fail, as it must.
cholcov([i 0 ;0 1])
ans =
[]
John D'Errico
el 15 de Feb. de 2015
@Betha - The answer to you is that IF this error has been returned, it is because no solution exists, or can exist. As I point out above, a matrix with complex diagonal elements cannot be factorized in the form
M = X*X'
Ver también
Categorías
Más información sobre Linear Algebra en Help Center y File Exchange.
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!