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Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA T will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A. An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix.
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A T (among other notations). [1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [2]
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
For the cases where has full row or column rank, and the inverse of the correlation matrix ( for with full row rank or for full column rank) is already known, the pseudoinverse for matrices related to can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the ...
Multiplying a matrix M by either or on either the left or the right will permute either the rows or columns of M by either π or π −1.The details are a bit tricky. To begin with, when we permute the entries of a vector (, …,) by some permutation π, we move the entry of the input vector into the () slot of the output vector.
In higher dimensions, however, there exist maps that can't be decomposed in this fashion, and the criterion is no longer sufficient. Consequently, there are entangled states which have a positive partial transpose. Such states have the interesting property that they are bound entangled, i.e. they can not be distilled for quantum communication ...
The self dot product of a complex vector =, involving the conjugate transpose of a row vector, is also known as the norm squared, = ‖ ‖, after the Euclidean norm; it is a vector generalization of the absolute square of a complex scalar (see also: Squared Euclidean distance).
A skew-symmetric graph is a graph that is isomorphic to its own transpose graph, via a special kind of isomorphism that pairs up all of the vertices. The converse relation of a binary relation is the relation that reverses the ordering of each pair of related objects. If the relation is interpreted as a directed graph, this is the same thing as ...