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Encyclopedia > Conjugate transpose

In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. The conjugate transpose is formally defined by Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

(A^*)_{i,j} = overline{A_{j,i}}

where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...


This definition can also be written as

A^* = {overline A}^{T}

where A^T ,! denotes the transpose and overline A ,! denotes the matrix with complex conjugated entries.


Other names for the conjugate transpose of a matrix are Hermitian conjugate, or tranjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:

Note that in some contexts A^* ,! can be used to denote the matrix with complex conjugated entries so care must be taken not to confuse notations. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ... Fig. ... In mathematics, and in particular linear algebra, the pseudoinverse of an matrix is a generalization of the inverse matrix . ...

Contents

Example

If


A=begin{bmatrix}3+i&5 2-2i&iend{bmatrix}


then


A^*=begin{bmatrix}3-i&2+2i 5&-iend{bmatrix}.


Basic remarks

If the entries of A are real, then A* coincides with the transpose AT of A. It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...


A square matrix A with entries aij is called

  • Hermitian or self-adjoint if A = A*, i.e., a_{ij}=a_{ji}^* ;
  • skew Hermitian or antihermitian if A = −A*, i.e., a_{ij}=-a_{ji}^{*} ;
  • normal if A*A = AA*.

Even if A is not square, the two matrices A*A and AA* are both Hermitian and in fact positive semi-definite matrices. A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ... A complex square matrix A is a normal matrix if where A* is the conjugate transpose of A. (If A is a real matrix, A*=AT and so it is normal if ATA = AAT.) All unitary, hermitian, and skew-hermitian matrices are normal. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...


The adjoint matrix A* should not be confused with the adjugate adj(A) (which is also sometimes called "adjoint"). In linear algebra, the adjugate or classical adjoint of a square matrix is a matrix which plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions. ...


Motivation

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 skew-symmetric matrices, obeying matrix addition and multiplication:

a + ib equiv Big(begin{matrix} a & -b  b & a end{matrix}Big)

An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n matrix of real numbers. It therefore arises very naturally that when transposing such a matrix which is made up of complex numbers, one may in the process also have to take the complex conjugate of each entry.


Properties of the conjugate transpose

  • (A + B)* = A* + B* for any two matrices A and B of the same dimensions.
  • (rA)* = r*A* for any complex number r and any matrix A. Here r* refers to the complex conjugate of r.
  • (AB)* = B*A* for any m-by-n matrix A and any n-by-p matrix B. Note that the order of the factors is reversed.
  • (A*)* = A for any matrix A.
  • If A is a square matrix, then det (A*) = (det A)* and trace (A*) = (trace A)*
  • A is invertible if and only if A* is invertible, and in that case we have (A*)-1 = (A-1)*.
  • The eigenvalues of A* are the complex conjugates of the eigenvalues of A.
  • <Ax,y> = <x, A*y> for any m-by-n matrix A, any vector x in Cn and any vector y in Cm. Here <.,.> denotes the ordinary Euclidean inner product (or dot product) on Cm and Cn.

In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

Generalizations

The last property given above shows that if one views A as a linear transformation from the Euclidean Hilbert space Cn to Cm, then the matrix A* corresponds to the adjoint operator of A. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ...


Another generalization is available: suppose A is a linear map from a complex vector space V to another W, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A. It maps the conjugate dual of W to the conjugate dual of V. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, one associates to every complex vector space V its complex conjugate vector space V*, again a complex vector space. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...


See also

  • Hermitian conjugate

In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ...

External links


  Results from FactBites:
 
Transpose (449 words)
Informally, the transpose of a square matrix is obtained by reflecting at the main diagonal (that runs from the top left to bottom right of the matrix).
This shows that the transpose is a linear map from the space of all m-by-n matrices to the space of all n-by-m matrices.
A square matrix whose transpose is equal to itself is called a symmetric matrix, i.e.
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