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Adobe Photoshop generates a pseudo-random noise dither pattern on startup, with each pixel location in a 2D raster array assigned a gray value (R=G=B) and an alpha value of 1 ("on"). As the opacity of the top layer is reduced, the alpha value of some of the gray pixels is switched from 1 to 0 ("off"), with the result that image pixels ...
A "cross number" Fill-In Another Fill-in variation [clarification needed]. A common variation on the standard Fill-In is using numbers, instead of specific words, sometimes called "cross numbers".
Matrix pencils play an important role in numerical linear algebra.The problem of finding the eigenvalues of a pencil is called the generalized eigenvalue problem.The most popular algorithm for this task is the QZ algorithm, which is an implicit version of the QR algorithm to solve the eigenvalue problem = without inverting the matrix (which is impossible when is singular, or numerically ...
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
It is also known as Principal Coordinates Analysis (PCoA), Torgerson Scaling or Torgerson–Gower scaling. It takes an input matrix giving dissimilarities between pairs of items and outputs a coordinate matrix whose configuration minimizes a loss function called strain, [2] which is given by (,,...,) = (, (),) /, where denote vectors in N-dimensional space, denotes the scalar product between ...
Matrix theory is a branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra , but soon grew to include subjects related to graph theory , algebra , combinatorics and statistics .
The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that: When multiplied by itself, the result is itself; All of its rows and columns are linearly independent. The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However ...