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The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. [3] [4] Computing matrix products is a central operation in all computational applications of linear algebra.
The straightforward multiplication of a matrix that is X × Y by a matrix that is Y × Z requires XYZ ordinary multiplications and X(Y − 1)Z ordinary additions. In this context, it is typical to use the number of ordinary multiplications as a measure of the runtime complexity. If A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × ...
The Hadamard product operates on identically shaped matrices and produces a third matrix of the same dimensions. In mathematics, the Hadamard product (also known as the element-wise product, entrywise product [1]: ch. 5 or Schur product [2]) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements.
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.
Two weighing matrices are considered to be equivalent if one can be obtained from the other by a series of permutations and negations of the rows and columns of the matrix. The classification of weighing matrices is complete for cases where w ≤ 5 {\displaystyle w\leq 5} as well as all cases where n ≤ 15 {\displaystyle n\leq 15} are also ...
This new matrix A 3 is the upper triangular matrix needed to perform an iteration of the QR decomposition. Q is now formed using the transpose of the rotation matrices in the following manner: Q = G 1 T G 2 T . {\displaystyle Q=G_{1}^{T}\,G_{2}^{T}.}
The entries in the matrix make clear the advantage of adding pseudocounts, especially when using small datasets to construct M. The background model need not have equal values for each symbol: for example, when studying organisms with a high GC-content , the values for C and G may be increased with a corresponding decrease for the A and T values.