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Kronecker product. In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a ...
In matrix inversion however, instead of vector b, we have matrix B, where B is an n-by-p matrix, so that we are trying to find a matrix X (also a n-by-p matrix): = =. We can use the same algorithm presented earlier to solve for each column of matrix X. Now suppose that B is the identity matrix of size n.
QR decomposition. In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) problem and is the basis for a particular ...
An n×n matrix with n distinct 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 VD1/2 V−1, where D1/2 is ...
Modular multiplicative inverse. In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. [1] In the standard notation of modular arithmetic this congruence is written as.
In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. [1] Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each .
For example, setting c = d = 0 produces a diagonal complex matrix representation of complex numbers, and setting b = d = 0 produces a real matrix representation. The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.
The following table shows how the extended Euclidean algorithm proceeds with input 240 and 46. The greatest common divisor is the last non zero entry, 2 in the column "remainder". The computation stops at row 6, because the remainder in it is 0. Bézout coefficients appear in the last two columns of the second-to-last row.