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To reference a particular element in a matrix, specify its row
and column number using the following syntax, where `A`

is
the matrix variable. Always specify the row first and column second:

A(row, column)

For example, for a 4-by-4 magic square `A`

,

A = magic(4) A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1

you would access the element at row 4, column 2 with

A(4,2) ans = 14

For arrays with more than two dimensions, specify additional
indices following the `row`

and `column`

indices.
See the section on Multidimensional Arrays.

You can refer to the elements of a MATLAB^{®} matrix with a
single subscript, `A(k)`

. MATLAB stores matrices
and arrays not in the shape that they appear when displayed in the MATLAB Command
Window, but as a single column of elements. This single column is
composed of all of the columns from the matrix, each appended to the
last.

So, matrix `A`

A = [2 6 9; 4 2 8; 3 5 1] A = 2 6 9 4 2 8 3 5 1

is actually stored in memory as the sequence

2, 4, 3, 6, 2, 5, 9, 8, 1

The element at row 3, column 2 of matrix `A`

(value
= `5`

) can also be identified as element 6 in the
actual storage sequence. To access this element, you have a choice
of using the standard `A(3,2)`

syntax, or you can
use `A(6)`

, which is referred to as *linear
indexing*.

If you supply more subscripts, MATLAB calculates an index
into the storage column based on the dimensions you assigned to the
array. For example, assume a two-dimensional array like `A`

has
size `[d1 d2]`

, where `d1`

is the
number of rows in the array and `d2`

is the number
of columns. If you supply two subscripts `(i, j)`

representing
row-column indices, the offset is

(j-1) * d1 + i

Given the expression `A(3,2)`

, MATLAB calculates
the offset into `A`

's storage column as ```
(2-1)
* 3 + 3
```

, or `6`

. Counting down six elements
in the column accesses the value `5`

.

If you have row-column subscripts but want to use linear indexing
instead, you can convert to the latter using the `sub2ind`

function. In the 3-by-3 matrix `A`

used
in the previous section, `sub2ind`

changes a standard
row-column index of (3,2) to a linear index of `6`

:

A = [2 6 9; 4 2 8; 3 5 1]; linearindex = sub2ind(size(A), 3, 2) linearindex = 6

To get the row-column equivalent of a linear index, use the `ind2sub`

function:

[row col] = ind2sub(size(A), 6) row = 3 col = 2

When you assign to elements outside the bounds of a numeric
array, MATLAB expands the array to include those elements and
fills the missing values with `0`

.

Assign a value to an element outside the bounds of `A`

.

A = magic(4); A(3,5) = 7 A = 16 2 3 13 0 5 11 10 8 0 9 7 6 12 7 4 14 15 1 0

When you extend structure and cell arrays, MATLAB fills
unaddressed elements with an empty value. MATLAB fills unaddressed
elements in categorical arrays with `<undefined>`

.
For datetime arrays, MATLAB fills unaddressed elements with NaT
(Not-a-Time).

If you try to refer to elements outside an array on the right side of an assignment statement, MATLAB throws an error.

test = A(7,7) Index exceeds matrix dimensions.

For the 4-by-4 matrix `A`

shown below, it is
possible to compute the sum of the elements in the fourth column of `A`

by
typing

A = magic(4); A(1,4) + A(2,4) + A(3,4) + A(4,4)

You can reduce the size of this expression using the colon operator. Subscript expressions involving colons refer to portions of a matrix. The expression

A(1:m, n)

refers to the elements in rows `1`

through `m`

of
column `n`

of matrix `A`

. Using
this notation, you can compute the sum of the fourth column of `A`

more
succinctly:

sum(A(1:4, 4))

To refer to nonconsecutive elements in a matrix, use the colon
operator with a step value. The `m:3:n`

in this expression
means to make the assignment to every third element in the matrix.
Note that this example uses linear indexing:

B = A; B(1:3:16) = -10 B = -10 2 3 -10 5 11 -10 8 9 -10 6 12 -10 14 15 -10

MATLAB supports a type of array indexing that uses one array as the index into another array. You can base this type of indexing on either the values or the positions of elements in the indexing array.

Here is an example of value-based indexing where array `B`

indexes
into elements `1`

, `3`

, `6`

, `7`

,
and `10`

of array `A`

. In this case,
the *numeric* *values* of array `B`

designate
the intended elements of `A`

:

A = 5:5:50 A = 5 10 15 20 25 30 35 40 45 50 B = [1 3 6 7 10]; A(B) ans = 5 15 30 35 50

If you index into a vector with another vector, the orientation of the indexed vector is honored for the output:

A(B') ans = 5 15 30 35 50 A1 = A'; A1(B) ans = 5 15 30 35 50

If you index into a vector with a nonvector, the shape of the indices is honored:

C = [1 3 6; 7 9 10]; A(C) ans = 5 15 30 35 45 50

MATLAB provides the keyword `end`

to
designate the last element in a particular dimension of an array.
This keyword can be useful in instances where your program does not
know how many rows or columns there are in a matrix. You can replace
the expression in the previous example with

B(1:3:end) = -10

The colon by itself refers to *all* the elements
in a row or column of a matrix. Using the following syntax, you can
compute the sum of all elements in the second column of a 4-by-4 magic
square `A`

:

sum(A(:, 2)) ans = 34

By using the colon with linear indexing, you can refer to all
elements in the entire matrix. This example displays all the elements
of matrix `A`

, returning them in a column-wise order:

A(:) ans = 16 5 9 4 . . . 12 1

A logical array index designates the elements of an array `A`

based
on their *position* in the indexing array, `B`

,
not their value. In this *masking* type of operation,
every `true`

element in the indexing array is treated
as a positional index into the array being accessed.

In the following example, `B`

is a matrix of
logical ones and zeros. The position of these elements in `B`

determines
which elements of `A`

are designated by the expression `A(B)`

:

A = [1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 9 B = logical([0 1 0; 1 0 1; 0 0 1]) B = 3×3 logical array 0 1 0 1 0 1 0 0 1 A(B) ans = 4 2 6 9

The `find`

function can
be useful with logical arrays as it returns the linear indices of
nonzero elements in `B`

, and thus helps to interpret `A(B)`

:

find(B) ans = 2 4 8 9

This example creates logical array `B`

that
satisfies the condition `A > 0.5`

, and uses the
positions of ones in `B`

to index into `A`

:

rng(0,'twister'); % Reset the random number generator A = rand(5); B = A > 0.5; A(B) = 0 A = 0 0.0975 0.1576 0.1419 0 0 0.2785 0 0.4218 0.0357 0.1270 0 0 0 0 0 0 0.4854 0 0 0 0 0 0 0

A simpler way to express this is

A(A > 0.5) = 0

The next example highlights the location of the prime numbers
in a magic square using logical indexing to set the nonprimes to `0`

:

A = magic(4) A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1 B = isprime(A) B = 4×4 logical array 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 A(~B) = 0; % Logical indexing A A = 0 2 3 13 5 11 0 0 0 7 0 0 0 0 0 0

find(B) ans = 2 5 6 7 9 13

In most cases, the logical indexing array should have the same number of elements as the array being indexed into, but this is not a requirement. The indexing array may have smaller (but not larger) dimensions:

A = [1 2 3;4 5 6;7 8 9] A = 1 2 3 4 5 6 7 8 9 B = logical([0 1 0; 1 0 1]) B = 2×3 logical array 0 1 0 1 0 1 isequal(numel(A), numel(B)) ans = logical 0 A(B) ans = 4 7 8

MATLAB treats the missing elements of the indexing array
as if they were present and set to zero, as in array `C`

below:

% Add zeros to indexing array C to give it the same number of % elements as A. C = logical([B(:);0;0;0]); isequal(numel(A), numel(C)) ans = logical 1 A(C) ans = 4 7 8

When you index into a standard MATLAB array using a single
colon, MATLAB returns a column vector (see variable `n`

,
below). When you index into a structure or cell array using a single
colon, you get a comma-separated list (see Access Data in a Structure Array and Access Data in Cell Array for more information.)

Create three types of arrays:

n = [1 2 3; 4 5 6]; c = {1 2; 3 4}; s = cell2struct(c, {'a', 'b'}, 1); s(:,2)=s(:,1);

Use single-colon indexing on each:

n(:) c{:} s(:).a ans = ans = ans = 1 1 1 4 ans = ans = 2 3 2 5 ans = ans = 3 2 1 6 ans = ans = 4 2

When assigning values from one matrix to another matrix, you can use any of the styles of indexing covered in this section. Matrix assignment statements also have the following requirement.

In the assignment `A(J,K,...) = B(M,N,...)`

,
subscripts `J`

, `K`

, `M`

, `N`

,
etc. may be scalar, vector, or array, provided that all of the following
are true:

The number of subscripts specified for

`B`

, not including trailing subscripts equal to 1, does not exceed`ndims`

`(B)`

.The number of nonscalar subscripts specified for

`A`

equals the number of nonscalar subscripts specified for`B`

. For example,`A(5, 1:4, 1, 2) = B(5:8)`

is valid because both sides of the equation use one nonscalar subscript.The order and length of all nonscalar subscripts specified for

`A`

matches the order and length of nonscalar subscripts specified for`B`

. For example,`A(1:4, 3, 3:9) = B(5:8, 1:7)`

is valid because both sides of the equation (ignoring the one scalar subscript`3`

) use a 4-element subscript followed by a 7-element subscript.