Documentation

mtimes, *

Matrix multiplication

Description

example

C = A*B is the matrix product of A and B. If A is an m-by-p and B is a p-by-n matrix, then C is an m-by-n matrix defined by

$C\left(i,j\right)=\sum _{k=1}^{p}A\left(i,k\right)B\left(k,j\right).$

This definition says that C(i,j) is the inner product of the ith row of A with the jth column of B. You can write this definition using the MATLAB® colon operator as

C(i,j) = A(i,:)*B(:,j)
For nonscalar A and B, the number of columns of A must equal the number of rows of B. Matrix multiplication is not universally commutative for nonscalar inputs. That is, A*B is typically not equal to B*A. If at least one input is scalar, then A*B is equivalent to A.*B and is commutative.

C = mtimes(A,B) is an alternative way to execute A*B, but is rarely used. It enables operator overloading for classes.

Examples

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Create a 1-by-4 row vector, A, and a 4-by-1 column vector, B.

A = [1 1 0 0];
B = [1; 2; 3; 4];

Multiply A times B.

C = A*B
C = 3

The result is a 1-by-1 scalar, also called the dot product or inner product of the vectors A and B. Alternatively, you can calculate the dot product $A\cdot B$ with the syntax dot(A,B).

Multiply B times A.

C = B*A
C = 4×4

1     1     0     0
2     2     0     0
3     3     0     0
4     4     0     0

The result is a 4-by-4 matrix, also called the outer product of the vectors A and B. The outer product of two vectors, $A\otimes B$, returns a matrix.

Create two arrays, A and B.

A = [1 3 5; 2 4 7];
B = [-5 8 11; 3 9 21; 4 0 8];

Calculate the product of A and B.

C = A*B
C = 2×3

24    35   114
30    52   162

Calculate the inner product of the second row of A and the third column of B.

A(2,:)*B(:,3)
ans = 162

This answer is the same as C(2,3).

Input Arguments

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Operands, specified as scalars, vectors, or matrices.

• A and B must be 2-D arrays, or one of them can be scalar.

• For nonscalar inputs, the number of columns in A must be equal to the number of rows in B.

• If one of A or B is an integer class (int16, uint8, …), then the other input must be a scalar. Operands with an integer data type cannot be complex.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char | duration | calendarDuration
Complex Number Support: Yes

Output Arguments

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Product, returned as a scalar, vector, or matrix. Array C has the same number of rows as input A and the same number of columns as input B. For example, if A is an m-by-0 empty matrix and B is a 0-by-n empty matrix, then A*B is an m-by-n matrix of zeros.

Tips

• With chained matrix multiplications such as A*B*C, you might be able to improve execution time by using parentheses to dictate the order of the operations. Consider the case of multiplying three matrices with A*B*C, where A is 500-by-2, B is 2-by-500, and C is 500-by-2.

• With no parentheses, the order of operations is left to right so A*B is calculated first, which forms a 500-by-500 matrix. This matrix is then multiplied with C to arrive at the 500-by-2 result.

• If you instead specify A*(B*C), then B*C is multiplied first, producing a 2-by-2 matrix. The small matrix then multiplies A to arrive at the same 500-by-2 result, but with fewer operations and less intermediate memory usage.