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Here, vec(X) denotes the vectorization of the matrix X, formed by stacking the columns of X into a single column vector. It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution, if and only if A and B are invertible ( Horn & Johnson 1991 , Lemma 4.3.1).
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
Note first that any 2 × 2 real matrix can be considered one of the three types of the complex number z = x + y ε, where ε 2 ∈ { −1, 0, +1 }. This z is a point on a complex subplane of the ring of matrices. [8] The case where the determinant is negative only arises in a plane with ε 2 =+1, that is a split-complex number plane. Only one ...
Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, [2] to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra , and as such has numerous applications in many areas of mathematics, as well as in applied ...
The system Q(Rx) = b is solved by Rx = Q T b = c, and the system Rx = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable .
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
[citation needed] The algorithms described below all involve about (1/3)n 3 FLOPs (n 3 /6 multiplications and the same number of additions) for real flavors and (4/3)n 3 FLOPs for complex flavors, [17] where n is the size of the matrix A. Hence, they have half the cost of the LU decomposition, which uses 2n 3 /3 FLOPs (see Trefethen and Bau 1997).
[1] [2] A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A . In the general linear group , similarity is therefore the same as conjugacy , and similar matrices are also called conjugate ; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive ...