In terms of computation, what is the difference between a diagonal matrix (for which the nonzero terms are in the diagonal: tridiagonal, pentadiagonal, ... and all the other elements are equal to zero) and a hollow matrix (for which almost all the elements are null exept a few ones?
Which is the best matrix to use for computation? Which one performs well for calculus?
In terms of computation all these matrices are relativity easy to handle however, the diagonal matrices are simple to generate and work with
Matrices Calculator is a lightweight and easy to use application that was designed in order to provide you with simple means of calculating matrices. It is able to solve several matrices, store them and more.
Rather than saying "hollow", the most common term is "sparse matrix". If you look for this in Wikipedia you'll find lots of information.
A diagonal matrix is easy to describe. It can be considered as a 1-dimensional array of numbers. e.g. Aij = 0 when i not equal to j, and Aii = dii. where dii is some number.
On the other hand, we cannot describe sparse matrix by an equation as is done above in case of diagonal matrix. The only well defined information about sparse matrix is that such a matrix has all most all entries having zero value with very few entries having a non-zero value.
Thanks to all, I now will return to the idea of diagonal matrix rather than sparse matrix. The matrices, I used are inversible, so I can convert them to diagonal matrices
Hello, Lakhdar Chiter !
Could you tell what kind of problem has lead you to the sparse and diagonal matrices?
In case you have to solve numerically a linear differential equation with initial and boundary conditions (a problem from the field of mathematical physics) and you decide to use the finite differences method, you will get a 3-diagonal matrix for a 1D problem, a 5-diagonal matrix for 2D problem and a 7-diagonal matrix for a 3D problem.
At the same time, if you decide to use finite elements method (instead of finite differences) you will most probably get a sparse matrix.
If you use Matlab in your computation, you will find that all these matrices are restored in the same way. For a banded or a sparse matrix, you may add several diagonal or subdiagonal matrices together to form it.
To Prof; Taras Dauzhenca;
Thank you for your reply and the question you advanced.
We have used Rationalized Haar functions to solve numerically some kinds of integral equations. In the computational method, we use the operational matrix of integration (and product). It appears that when the approximative solution is close to the exact solution, the obtained matrix is a sparse matrix (the number of null terms is up to 95). From this simple observation, I just thinked to diagonal matrix instead to any sparse matrix, and I will guess if the computational method will give any result. This is what I am going to see next. Thank you for your considerable help.
Hello Lakhar. I have no idea of the application you're describing, but I'll answer purely in terms of the fundamental characteristics of these matrices.
First, they are all sparse, and therefore efficient to represent on a computer and/or to perform forward arithmetic (add,subtract,multiply).
However, that is where the resemblance ends. Even a tridiagonal matrix needn't be invertible, and calculating its eigenvalues can involve a lot of calculation if it is very large. It may have large eigenspaces. The same is true of sparse matrices in general.
A diagonal matrix, however, is immediately seen to be invertible: If all of its diagonal entries are nonzero, that is, and its inverse is obtained by replacing them with their reciprocals. There is no such easy rule for sparse matrices in general. Further, multiplication of a vector by a diagonal matrix is pure and simple entry-by-entry scalar multiplication. Its eigenvalues are equal to the diagonal entries, and the standard basis is also a basis of eigenvectors.
There is a reason why in many applications the first thing one does is diagonalize the matrix: Once diagonalized, all arithmetic and vector geometry involving a matrix is trivial. Quite often, the solution to a nasty problem is as follows:
1. Express the problem in terms of a matrix
2. Perform a conjugation that diagonalizes that matrix (and also conjugate all vectors expressing elements of the problem).
3. Write down the solution for the system in diagonalized form. I say "write down" because it is generally immediate; there are no big intermediate steps here.
4. Reverse the conjugation to "lift" the solution back into its original form
5. Re-express the solution in the original context of the problem.
But deep complexity may be hidden in other matrices, even if they are very sparse.
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