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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
A matrix B is said to be a square root of A if the matrix product BB is equal to A. [1] Some authors use the name square root or the notation A 1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = B T B = A (for real-valued matrices, where B T is the ...
Consider the system of equations x + y + 2z = 3, x + y + z = 1, 2x + 2y + 2z = 2.. The coefficient matrix is = [], and the augmented matrix is (|) = [].Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.
This is because any two bases are related by an invertible matrix (the change of basis matrix), so one basis is unisolvent if and only if any other basis is unisolvent. Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem.
If the latter holds, then the solution is unique if and only if has full column rank, in which case + is a zero matrix. If solutions exist but A {\displaystyle A} does not have full column rank, then we have an indeterminate system , all of whose infinitude of solutions are given by this last equation.
The Lyapunov equation is linear; therefore, if contains entries, the equation can be solved in () time using standard matrix factorization methods.. However, specialized algorithms are available which can yield solutions much quicker owing to the specific structure of the Lyapunov equation.
Given matrices and , the Sylvester equation + = has a unique solution for any if and only if and do not share any eigenvalue. Proof. The equation A X + X B = C {\displaystyle AX+XB=C} is a linear system with m n {\displaystyle mn} unknowns and the same number of equations.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.