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A square matrix may have a multiplicative inverse, called an inverse matrix. In the common case where the entries belong to a commutative ring R, a matrix has an inverse if and only if its determinant has a multiplicative inverse in R. The determinant of a product of square matrices is the product of the determinants of the factors.
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:
Nilpotent matrix: A square matrix satisfying A q = 0 for some positive integer q. Equivalently, the only eigenvalue of A is 0. Normal matrix: A square matrix that commutes with its conjugate transpose: AA ∗ = A ∗ A: They are the matrices to which the spectral theorem applies. Orthogonal matrix: A matrix whose inverse is equal to its ...
The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science. As of January 2024, the best bound on the asymptotic complexity of a matrix multiplication algorithm is O(n 2.371552).
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication:
The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R. [2] The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1. When multiplied by another matrix, it isolates a specific row or column in arbitrary position.
Alabama athletics director Greg Byrne makes a case for the Crimson Tide to make the College Football Playoff despite not winning a conference championship and having three losses.
This reduces the number of matrix additions and subtractions from 18 to 15. The number of matrix multiplications is still 7, and the asymptotic complexity is the same. [6] The algorithm was further optimised in 2017, [7] reducing the number of matrix additions per step to 12 while maintaining the number of matrix multiplications, and again in ...