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numpy.org. NumPy (pronounced / ˈnʌmpaɪ / 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 ] The predecessor of NumPy, Numeric, was originally created by Jim Hugunin with ...
For finding all the roots, arguably the most reliable method is the Francis QR algorithm computing the eigenvalues of the companion matrix corresponding to the polynomial, implemented as the standard method [1] in MATLAB. The oldest method of finding all roots is to start by finding a single root. When a root r has been found, it can be removed ...
The characteristic equation, also known as the determinantal equation, [ 1 ][ 2 ][ 3 ] is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. [ 4 ]
An n×n matrix with ndistinct nonzero eigenvalues has 2 n square roots. Such a matrix, A, has an eigendecomposition VDV−1 where V is the matrix whose columns are eigenvectors of A and D is the diagonal matrix whose diagonal elements are the corresponding n eigenvalues λi. Thus the square roots of A are given by VD1/2V−1, where D1/2 is any ...
Trace (linear algebra) In linear algebra, the trace of a square matrix A, denoted tr (A), [1] is the sum of the elements on its main diagonal, . It is only defined for a square matrix (n × n). In mathematical physics, if tr (A) = 0, the matrix is said to be traceless. This misnomer is widely used, as in the definition of Pauli matrices.
Each Jordan block contains one number lambda on its main diagonal, and ones above the main diagonal. The lambdas are the eigenvalues of the matrix; they need not be distinct. In linear algebra, a Jordan normal form, also known as a Jordan canonical form, [ 1 ][ 2 ] is an upper triangular matrix of a particular form called a Jordan matrix ...
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial. this polynomial evaluated at a matrix is. where is the identity matrix. [1] Note that has the same dimension as . A matrix polynomial equation is an equality between two matrix polynomials, which holds for the ...
Vandermonde matrix. In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix. The determinant of a square Vandermonde matrix (when ) is called a Vandermonde determinant or Vandermonde polynomial. Its value is: