Search results
Results from the WOW.Com Content Network
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 ...
Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b both have integer components and M is unimodular, has an integer solution.
Non-square matrices, i.e. m-by-n matrices for which m ≠ n, do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = I n.
Cramer's rule is an explicit formula for the solution of a system of linear equations, ... (if any exist) are given using the Moore–Penrose inverse of A, ...
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or {1,2,5}-inverse and denoted A #. The group inverse can be defined, equivalently, by the properties AA # A = A, A # AA # = A #, and AA # = A # A. A projection matrix P, defined as a matrix such that P 2 = P, has index 1 (or 0) and has Drazin inverse P D = P.
This formula also has application in theoretical physics. Namely, in quantum field theory, one uses this formula to calculate the propagator of a spin-1 field. [8] [circular reference] The inverse propagator (as it appears in the Lagrangian) has the form +.