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By contrast, if most of the elements are non-zero, the matrix is considered dense. [1] The number of zero-valued elements divided by the total number of elements (e.g., m × n for an m × n matrix) is sometimes referred to as the sparsity of the matrix. Conceptually, sparsity corresponds to systems with few pairwise interactions.
Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m × n matrix and B is an n × p matrix, then their matrix product AB is the m × p matrix whose entries are given by dot product of the corresponding row of A and the corresponding ...
In mathematics, an anti-diagonal matrix is a square matrix where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal or bad diagonal).
Therefore, det(R – I) = 0, meaning there is a nonzero vector v with (R – I)v = 0, that is Rv = v, a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to v , so the total dimension of fixed eigenvectors is odd.
The non-zero singular values of (found on the diagonal entries of ) are the square roots of the non-zero eigenvalues of both and . The SVD of a matrix M {\displaystyle \mathbf {M} } is typically computed by a two-step procedure.
So there exists an invertible matrix P such that P −1 AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal. J is called the Jordan normal form of A. Each J i is called a Jordan block of A. In a given Jordan block, every entry on the superdiagonal is 1. Assuming this result, we can deduce the following ...
NumPy (pronounced / ˈ n ʌ m p aɪ / NUM-py) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. [3]
Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either strictly diagonally dominant, [1] or symmetric and positive definite. It was only mentioned in a private letter from Gauss to his student Gerling in 1823. [2] A publication was not delivered before 1874 by ...