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Concept# Transpose

Summary

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 AT (among other notations).
The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT.
The transpose of a matrix A, denoted by AT, ^⊤A, A^⊤, , A′, Atr, tA or At, may be constructed by any one of the following methods:
Reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT
Write the rows of A as the columns of AT
Write the columns of A as the rows of AT
Formally, the i-th row, j-th column element of AT is the j-th row, i-th column element of A:
If A is an m × n matrix, then AT is an n × m matrix.
In the case of square matrices, AT may also denote the Tth power of the matrix A. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as TA. An advantage of this notation is that no parentheses are needed when exponents are involved: as (^TA)^n = ^T(A^n), notation ^TA^n is not ambiguous.
In this article this confusion is avoided by never using the symbol T as a variable name.
A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if
A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if
A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if
A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if
A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, A is unitary if
Let A and B be matrices and c be a scalar.

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In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

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In linear algebra, an invertible complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if where I is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

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