Matrix Multiplication Rules
Here are some important rules and properties of Matrix Multiplication Rules:
Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. In order to multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
If the dimensions of the matrices are A (m x n) and B (n x p), then the resulting matrix C will be of dimensions (m x p).
Table of Contents
Associativity
Matrix multiplication is associative, meaning that the order in which you multiply three matrices doesn’t affect the result. Mathematically, (A⋅B)⋅C=A⋅(B⋅C)
This property allows us to write expressions without parentheses, and it doesn’t matter how the multiplication is grouped.
Distributivity
Matrix multiplication is distributive over addition. This property resembles the distributive property in arithmetic and is useful for simplifying expressions involving matrices.
Mathematically, A⋅(B+C)=A⋅B+A⋅C.
Not Commutative
Matrix multiplication is not commutative, meaning that the order of multiplication matters. In general, A⋅B is not necessarily equal to B⋅A. This property distinguishes matrix multiplication from scalar multiplication.
Identity Matrix
If I is the identity matrix of appropriate size, then multiplying any matrix A by the identity matrix does not change A. Mathematically, A⋅I=A and I⋅A=A. The identity matrix acts as a neutral element in matrix multiplication.
Zero Matrix
If O is the zero matrix of appropriate size, then multiplying any matrix A by the zero matrix results in the zero matrix. Mathematically, A⋅O=O and O⋅A=O.
This property highlights the role of the zero matrix as an “annihilator” in matrix multiplication.
Scalar Multiplication
Multiplying a matrix A by a scalar c involves multiplying each element of A by c. If A has dimensions m×n, then c⋅A is also m×n.
Transpose of a Product
The transpose of the product of two matrices is equal to the product of their transposes in reverse order. Mathematically, (A⋅B)T=BT⋅AT
Inverse of a Product
The inverse of the product of two invertible matrices is equal to the product of their inverses in reverse order. If A and B are invertible matrices, then (A⋅B)−1=B−1⋅A−1
Block Matrix Multiplication
Matrices can be partitioned into blocks, and matrix multiplication can be performed on these blocks. The resulting block matrix has elements obtained from the block-wise products.
Trace of a Product
The trace of the product of two square matrices is equal to the trace of their reverse product. That is, Tr(A⋅B)=Tr(B⋅A), where Tr denotes the trace of a matrix (sum of diagonal elements).
Kronecker Product
The Kronecker product, denoted by ⊗, is a generalization of matrix multiplication that combines two matrices to form a larger matrix. The Kronecker product has various applications in linear algebra and signal processing.
Einstein Summation Convention
This convention simplifies expressions involving matrices and vectors by implying summation over repeated indices. It is often used in the context of tensor notation, which extends matrix notation to higher-dimensional arrays.
Matrix Exponentiation
Matrix exponentiation involves raising a square matrix to a power. It is used in various mathematical and scientific contexts, including solving linear systems and differential equations.
Schur Product (Hadamard Product)
The Schur product of two matrices, denoted by ⊙, involves multiplying corresponding elements of two matrices to produce a new matrix. It is also known as the Hadamard product and is different from standard matrix multiplication.
Matrix Division
Unlike matrix multiplication, matrix division is not a straightforward operation. Instead, it is often approached through the concept of matrix inversion.