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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:
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
Multiplying a row by a number multiplies the determinant by this number. Adding a multiple of one row to another row does not change the determinant. The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns. The determinant is invariant under matrix similarity.
Using a naive lower bound and schoolbook matrix multiplication for the upper bound, one can straightforwardly conclude that 2 ≤ ω ≤ 3. Whether ω = 2 is a major open question in theoretical computer science, and there is a line of research developing matrix multiplication algorithms to get improved bounds on ω.
A bilinear map is a function: such that for all , the map (,) is a linear map from to , and for all , the map (,) is a linear map from to . In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.
the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in the space of solutions, if it is not empty; the co-kernel is the space of constraints that the solutions must satisfy, and its dimension is the maximal number of independent constraints.
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 × ...
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.