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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 ...
If is a singular matrix of rank , then it admits an LU factorization if the first leading principal minors are nonzero, although the converse is not true. [9] If a square, invertible matrix has an LDU (factorization with all diagonal entries of L and U equal to 1), then the factorization is unique. [8]
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.
Unit-Scale-Invariant Singular-Value Decomposition: =, where S is a unique nonnegative diagonal matrix of scale-invariant singular values, U and V are unitary matrices, is the conjugate transpose of V, and positive diagonal matrices D and E.
So there is a unique solution to the original system of equations. Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in reduced row echelon form, as it is done in the table. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss–Jordan elimination, to distinguish ...
The solution is the product . [3] This intuitively makes sense because an orthogonal matrix would have the decomposition where is the identity matrix, so that if = then the product = amounts to replacing the singular values with ones.
If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A −1. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or ...
P is the unknown n by n symmetric matrix and A, B, Q, R are known real coefficient matrices, with Q and R symmetric. Though generally this equation can have many solutions, it is usually specified that we want to obtain the unique stabilizing solution, if such a solution exists.
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