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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 identity that characterizes the transpose, that is, [(),] = [, ()], is formally similar to the definition of the Hermitian adjoint, however, the transpose and the Hermitian adjoint are not the same map.
The identity is also a permutation matrix. A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.
The only non-singular idempotent matrix is the identity matrix; that is, ... where superscript T indicates a transpose, and the vector of residuals is [2]
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.
Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution. The definition of involution extends readily to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f 2 is the identity homomorphism on M.
We denote the n×n identity matrix by I and the zero matrix by 0. The matrix exponential satisfies the following properties. [2] We begin with the properties that are immediate consequences of the definition as a power series: e 0 = I; exp(X T) = (exp X) T, where X T denotes the transpose of X. exp(X ∗) = (exp X) ∗, where X ∗ denotes the ...
Here is the conjugate transpose of V (or simply the transpose, if V contains real numbers only), and I denotes the identity matrix (of some dimension). Comment: The diagonal elements of D are called the singular values of A .