Matrix Operations vs Elementwise Operations

Concept of Matrix Operations

In MATLAB, many operations are defined in terms of linear algebra. When you write expressions with matrices and vectors, MATLAB often interprets them as matrix operations from linear algebra, not as simple independent operations on each entry.

A matrix operation treats an array as a single mathematical object. For example, matrix multiplication combines rows of one matrix with columns of another based on the rules of linear algebra. This is very different from taking two arrays of the same size and multiplying corresponding elements.

Matrix operations are available only when the sizes of the matrices satisfy certain algebraic rules. If the dimensions are incompatible for a particular matrix operation, MATLAB produces an error, even if the arrays have the same number of elements.

Matrix Multiplication

The most important matrix operation is matrix multiplication. In MATLAB, the symbol for matrix multiplication is . If you have a matrix $A$ of size $m \times n$ and a matrix $B$ of size $n \times p$, then the product $C = A B$ is defined and has size $m \times p$.

Entry $c_{ij}$ of the result $C$ is computed as

$$
c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}.
$$

This formula means each element of the result is a sum of products along a row of $A$ and a column of $B$. Notice that you do not multiply entries one by one in the same position. Instead, you combine entire rows with entire columns.

Matrix multiplication requires that the number of columns of the first matrix is equal to the number of rows of the second matrix. If you try to compute A * B and the inner dimensions do not match, MATLAB displays an error that mentions a mismatch in inner dimensions.

Scalar multiplication is a special case, but still uses . If s is a scalar and A is a matrix, then s A and A * s both multiply every element of A by s. Here the operation follows the usual linear algebra definition of scaling a matrix by a scalar.

Matrix Powers

MATLAB uses ^ for matrix powers. If $A$ is a square matrix and $n$ is a positive integer, then A^n means repeated matrix multiplication

$$
A^n = \underbrace{A A \dots * A}_{n \text{ times}}.
$$

The base must be a square matrix for ^ to make sense as a matrix power. MATLAB checks that the matrix is square, and if not, it returns an error.

Matrix powers with negative or noninteger exponents are also interpreted in a linear algebra sense when possible. For example, A^(-1) is the matrix inverse of A if A is invertible. Such uses depend on other linear algebra concepts and functions that are covered elsewhere, but it is important to see that ^ on a matrix is not just a simple elementwise squaring.

Matrix Division

MATLAB defines matrix division by \ and /. These symbols correspond to solving certain matrix equations in linear algebra, not to dividing individual entries.

The expression A \ B solves a linear system of the form

$$
A X = B
$$

and returns X. The expression A / B is related to solving

$$
X B = A.
$$

Both \ and / use matrix algebra rules and rely on properties of the underlying matrices. They do not perform entry by entry division. The detailed use of matrix division for solving linear systems is described in linear algebra topics, but here you should recognize that \ and / without dots are matrix operations, not elementwise ones.

Elementwise Operations

Elementwise operations treat arrays as collections of independent numbers. Instead of applying linear algebra rules, MATLAB applies the same arithmetic operation separately to each pair of corresponding elements.

Elementwise operations require that the arrays are compatible in size for element-by-element computation. For most beginner cases, that means the arrays have exactly the same size. When this is true, MATLAB produces a result array of the same size, where each entry is formed from the entries in the same position in the input arrays.

Elementwise operators in MATLAB are written using a dot before the symbol. The main elementwise operators are:

.* for elementwise multiplication
./ for elementwise division
.\ for elementwise left division
.^ for elementwise powers

If A and B are the same size, then A .* B produces an array whose $(i,j)$ entry is simply $a_{ij} b_{ij}$. No sums or combinations of rows and columns are involved.

Elementwise operations are often described as acting component by component. They are very useful when you want to apply the same scalar operation independently at each array location without any linear algebra meaning.

Elementwise Powers

Elementwise powers with . ^ are very common. The expression A .^ 2 squares each entry of A individually. If $A$ has entry $a_{ij}$ in position $(i,j)$, then the result has entry

$$
(a_{ij})^2
$$

in the same position. Unlike A^2, which is a matrix power and requires a square matrix, A .^ 2 works for any array shape, as long as it is numeric and the exponent makes sense.

This distinction between ^ and . ^ is essential. One is a matrix operation based on repeated matrix multiplication, the other is a simple elementwise arithmetic operation.

Elementwise Multiplication and Division

When you want to multiply or divide two arrays entry by entry, you must use the dotted forms ., ./, or .\. For example, if X and Y are the same size, then X . Y creates a new array whose entries are

$$
(x_{ij}) (y_{ij})
$$

for each location $(i,j)$.

Elementwise division uses ./ or .\. For X ./ Y, MATLAB computes

$$
\frac{x_{ij}}{y_{ij}}
$$

for each entry, as long as the denominator entry is not zero. The operator .\ reverses the order in each fraction, so X .\ Y gives

$$
\frac{y_{ij}}{x_{ij}}
$$

for each entry. In practice, ./ is much more commonly used.

Elementwise division is different from A \ B or A / B without dots. The dotted versions do not attempt to solve any linear systems. They simply divide corresponding entries.

Size Compatibility and Broadcasting

Both matrix and elementwise operations require some form of size compatibility, but the rules differ.

For matrix multiplication A * B, only the inner dimensions must match. That is, if A is of size m x n and B is n x p, the product is defined and has size m x p. There is no requirement that the outer dimensions be equal.

For elementwise operations such as A .* B, the arrays usually need to have the same size. When that is true, MATLAB combines elements at the same positions. If the sizes do not match, MATLAB either gives an error or, in some situations, uses broadcasting rules, where one array is implicitly expanded along a singleton dimension. This automatic expansion allows expressions such as multiplying a column vector by a row vector element by element, but these cases depend on the specific sizes of the arrays.

Even with broadcasting, the conceptual idea remains that each result entry is formed from one entry in each input array at that same logical position. There are no sums over rows and columns in elementwise operations.

Choosing Between Matrix and Elementwise Operations

Choosing the correct operator depends on what you want to compute. If you are working with a mathematical model that explicitly uses linear algebra constructs, such as matrix products or solutions of systems of equations, then you should expect to use matrix operators like *, ^, \, and / without dots.

If you are applying the same numeric transformation independently to each position in your data, you should use the dotted operators .*, ./, .\, and . ^. These let you write compact code that processes entire arrays at once, with each element treated separately.

Paying attention to the presence or absence of the dot is critical. The meanings of and ., or ^ and . ^, are fundamentally different. Using the wrong form can either produce an error about incompatible sizes or, worse, produce a result that is syntactically valid but mathematically incorrect for your intention.