Dictionary and translator for handheld
New : sensagent is now available on your handheld
A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !
With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.
Improve your site content
Add new content to your site from Sensagent by XML.
Crawl products or adds
Get XML access to reach the best products.
Index images and define metadata
Get XML access to fix the meaning of your metadata.
Please, email us to describe your idea.
Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, 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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by
where the subscripts denote the i,j-th entry, for 1 ≤ i ≤ n and 1 ≤ j ≤ m, and the overbar denotes a scalar complex conjugate. (The complex conjugate of , where a and b are reals, is .)
This definition can also be written as
where denotes the transpose and denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian conjugate, or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:
In some contexts, denotes the matrix with complex conjugated entries, and thus the conjugate transpose is denoted by or .
A square matrix A with entries is called
Even if A is not square, the two matrices A*A and AA* are both Hermitian and in fact positive semi-definite matrices.
The adjoint matrix A* should not be confused with the adjugate adj(A) (which is also sometimes called "adjoint").
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:
That is, denoting each complex number z by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space ) affected by complex z-multiplication on .
An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n matrix of real numbers. The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix, when viewed back again as n-by-m matrix made up of complex numbers.
The last property given above shows that if one views A as a linear transformation from the Euclidean Hilbert space to , 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.
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.