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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.
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.
More precisely, the sign of the elementary product needed to calculate the determinant of an anti-diagonal matrix is related to whether the corresponding triangular number is even or odd. This is because the number of inversions in the permutation for the only nonzero signed elementary product of any n × n anti-diagonal matrix is always equal ...
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 ...
The determinant of A is nonzero: det A ≠ 0. (In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit (i.e. multiplicatively invertible element) of that ring. The number 0 is not an eigenvalue of A.
For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation.
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]
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...