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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 ...
In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
A symmetric matrix can always be transformed in this way into a diagonal matrix which has only entries , + , along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A {\displaystyle A} , i.e. it does not depend on the matrix S {\displaystyle S} used.
Several important classes of matrices are subsets of each other. This article lists some important classes of matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries. Matrices have a long history of both study and application, leading to ...
In general, given some linear map f : V → V (where V is a finite-dimensional vector space), we can define the trace of this map by considering the trace of a matrix representation of f, that is, choosing a basis for V and describing f as a matrix relative to this basis, and taking the trace of this square matrix.
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.
These matrices are related as follows. The following statements are equivalent: A is similar over F to (), i.e. A can be conjugated to its companion matrix by matrices in GL n (F); the characteristic polynomial () coincides with the minimal polynomial of A, i.e. the minimal polynomial has degree n;
Matrix multiplication also does not necessarily obey the cancellation law. If AB = AC and A ≠ 0, then one must show that matrix A is invertible (i.e. has det(A) ≠ 0) before one can conclude that B = C. If det(A) = 0, then B might not equal C, because the matrix equation AX = B will not have a unique solution for a non-invertible matrix A.