How can we find inverse of a matrix using Simplex method? Do we need to have square matrix only or inverse can be found of any matrix? Also specify about the upper bound on the matrix size?
The Matrix Inverse is required during simplex only over the Basis Matrix (Basis Inversion).
Base matrix is a square matrix of dimensions (mxm) where m is the total number of constraints.
This Matrix Inversion is carried out using either the Product form of Inverse or LU Decomposition.
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I want to create an orthogonal symmetric matrix of size 5 and size 6. Is there a way for me to generate this type of matrix? An orthogonal symmetric matrix is when a matrix A is equal to its transpose and also its inverse.
I've tried searching for ways to do this but all I get is how to generate an orthogonal matrix or how to generate a symmetric matrix. Couldn't find a way to generate an orthogonal symmetric matrix.
Can a 2D transpose convolution operation be represented as a matrix multiplication with the Toeplitz matrix, as can be done for a normal convolution?
I want to generalise some ideas from a dense network to a convolutional network. For normal convolutions, this is not a problem, as they can be represented as matrix multiplications with the Toeplitz matrix. But I couldn't find a clear mathematical formulation of transposed convolution, so I am not sure about this case.
I was looking for a mathematical answer, so should have probably asked somewhere else, anyways I think my latex write-up is correct and answers the question:
formula transposed convolution
I have a square matrix that is largely zeros but interspersed with values. Is there a way to 'solve' this matrix so that all the information in it is contained in its lower triangle only, with the upper triangle containing only zeros?
Not in general.
If symmetric and positive definite you can do a Cholesky Decomposition.
If non-symmetric you can do an LU decomposition.
The Q matrix in quadratic forms (x'Qx) can be made symmetric and then lower-triangular. This is sometimes used when setting up a Quadratic Programming (QP) model.
Sparse versions of decomposition approaches are a well-studied area (not trivial though). In large-scale LP solvers sparse LU (simplex) or sparse Cholesky (interior point) are widely used.
Discr. convolution can be represented as multiplication of input with matrix M.
Where M is presented a special case of Toeplitz matrices - circulant matrices.
The questions is: is 2d convolution can also be represented as matrix multiplication?
p.s. By dicr. convolution I mean dicr. convolution with indexing discrete samples in modulus fashion, i.e. the discrete signal is repeating
....X[n-1]x[0]x[1]...x[N-1]x[0]...
Yes, it can, but it will generally be a rather big matrix. If your data set is on a grid of size NxM, then the convolution is a matrix operating on a vector of length N*M; the convolution matrix has N2M2 elements.
If your convolution kernel is small, then the matrix will typically a band matrix where the width of the band is at least N or M.
I have to find the determinant of a symmetric square NxN matrix with M diagonals and M << N. Is there a more fast method than LU-decomposing the matrix?
Yes, there are special methods for band(ed) matrices that solve elimination with O(N*M^2) complexity. Arbitrary found article of Jeff Thorson