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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Matrix (mathematics)
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.
Unitary matrix
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.
Matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, , adding two matrices would have the geometric effect of applying each matrix transformation separately onto , then adding the transformed vectors. However, there are other operations that could also be considered addition for matrices, such as the direct sum and the Kronecker sum. Two matrices must have an equal number of rows and columns to be added.
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