We know that it is possible to write a matrix by LU decomposition as a product of an upper triangular matrix in a lower triangular matrix. But now my question is, is it possible to get just one diagonal matrix equivalent to any independent or dependent matrix? In addition, we know that with the stepwise row method, a matrix can be converted to upper triangular or lower triangular. But if we continue this process, can we get a diagonal matrix? Or is there another method or not? Thanks
The question is not clear for me. If you mean that you want to decompose a matrix A to LU with L or U being diagonal matrix, it is not possible in general. On the other hand, with a complete Gaussian elimination on augmented matrix [A I], I=identity matrix with the same size of A, you can find [I A^{-1}], the inverse of matrix A.
My question in the field of linear algebra and whether it is possible to find an equivalent diagonal matrix for any independent or dependent matrix? In other words, is it possible to find a diagonal matrix for each assumed matrix so that a one-to-one relationship can be defined between the two?
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