Table of Contents
What is matrix multiplication?
Matrix Multiplication In linear algebra, matrices play an important role in dealing with different concepts. A matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns in mathematics. We can perform various operations on matrices such as addition, subtraction, multiplication and so on.
How do you multiply two matrices?
You can multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix. (Link on columns vs rows ) In the picture above , the matrices can be multiplied since the number of columns in the 1st one, matrix A, equals the number of rows in the 2 nd, matrix B.
Is the multiplication of two matrices commutative?
The matrix multiplication is not commutative. In matrix multiplication, the order matters a lot. This shows that the matrix AB ≠BA. Hence, the multiplication of two matrices is not commutative. If A, B and C are the three matrices, the associative property of matrix multiplication states that,
Directly applying the mathematical definition of matrix multiplication gives an algorithm that takes time on the order of n3 field operations to multiply two n × n matrices over that field ( Θ (n3) in big O notation ).
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What is the asymptotic complexity of the matrix multiplication algorithm?
, the matrix multiplication algorithm with best asymptotic complexity runs in O (n2.3728596) time, given by Josh Alman and Virginia Vassilevska Williams, however this algorithm is a galactic algorithm because of the large constants and cannot be realized practically.
How many multiplications does it take to multiply 2-matrices?
It is based on a way of multiplying two 2 × 2 -matrices which requires only 7 multiplications (instead of the usual 8), at the expense of several additional addition and subtraction operations. Applying this recursively gives an algorithm with a multiplicative cost of