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Fig.1. Google matrix of Wikipedia articles network, written in the bases of PageRank index; fragment of top 200 X 200 matrix elements is shown, total size N=3282257 (from [1]) A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links between pages.
where is the identity matrix and P is a () matrix. When the generator matrix is in standard form, the code C is systematic in its first k coordinate positions. [3] A generator matrix can be used to construct the parity check matrix for a code
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
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 ...
A matrix H representing a linear function : whose kernel is C is called a check matrix of C (or sometimes a parity check matrix). Equivalently, H is a matrix whose null space is C . If C is a code with a generating matrix G in standard form, G = [ I k ∣ P ] {\displaystyle {\boldsymbol {G}}=[I_{k}\mid P]} , then H = [ − P T ∣ I n − k ...
The matrix exponential of another matrix (matrix-matrix exponential), [24] is defined as = = for any normal and non-singular n×n matrix X, and any complex n×n matrix Y. For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y , because the multiplication operator for matrix ...
A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2.If a self-dual code is such that each codeword's weight is a multiple of some constant >, then it is of one of the following four types: [1]