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Creating Multidimensional Arrays Accessing Multidimensional Array Properties Indexing Multidimensional Arrays Reshaping Multidimensional Arrays Computing with Multidimensional Arrays 
An array having more than two dimensions is called a multidimensional array in the MATLAB^{®} application. Multidimensional arrays in MATLAB are an extension of the normal twodimensional matrix. Matrices have two dimensions: the row dimension and the column dimension.
You can access a twodimensional matrix element with two subscripts: the first representing the row index, and the second representing the column index.
Multidimensional arrays use additional subscripts for indexing. A threedimensional array, for example, uses three subscripts:
The first references array dimension 1, the row.
The second references dimension 2, the column.
The third references dimension 3. This illustration uses the concept of a page to represent dimensions 3 and higher.
To access the element in the second row, third column of page 2, for example, you use the subscripts (2,3,2).
As you add dimensions to an array, you also add subscripts. A fourdimensional array, for example, has four subscripts. The first two reference a rowcolumn pair; the second two access the third and fourth dimensions of data.
Most of the operations that you can perform on matrices (i.e., twodimensional arrays) can also be done on multidimensional arrays.
You can use the same techniques to create multidimensional arrays that you use for twodimensional matrices. In addition, MATLAB provides a special concatenation function that is useful for building multidimensional arrays.
This section discusses
One way to create a multidimensional array is to create a twodimensional array and extend it. For example, begin with a simple twodimensional array A.
A = [5 7 8; 0 1 9; 4 3 6];
A is a 3by3 array, that is, its row dimension is 3 and its column dimension is 3. To add a third dimension to A,
A(:,:,2) = [1 0 4; 3 5 6; 9 8 7]
MATLAB responds with
A(:,:,1) = 5 7 8 0 1 9 4 3 6 A(:,:,2) = 1 0 4 3 5 6 9 8 7
You can continue to add rows, columns, or pages to the array using similar assignment statements.
To extend A in any dimension:
Increment or add the appropriate subscript and assign the desired values.
Assign the same number of elements to corresponding array dimensions. For numeric arrays, all rows must have the same number of elements, all pages must have the same number of rows and columns, and so on.
You can take advantage of the MATLAB scalar expansion capabilities, together with the colon operator, to fill an entire dimension with a single value:
A(:,:,3) = 5; A(:,:,3) ans = 5 5 5 5 5 5 5 5 5
To turn A into a 3by3by3by2, fourdimensional array, enter
A(:,:,1,2) = [1 2 3; 4 5 6; 7 8 9]; A(:,:,2,2) = [9 8 7; 6 5 4; 3 2 1]; A(:,:,3,2) = [1 0 1; 1 1 0; 0 1 1];
Note that after the first two assignments MATLAB pads A with zeros, as needed, to maintain the corresponding sizes of dimensions.
You can use MATLAB functions such as randn, ones, and zeros to generate multidimensional arrays in the same way you use them for twodimensional arrays. Each argument you supply represents the size of the corresponding dimension in the resulting array. For example, to create a 4by3by2 array of normally distributed random numbers:
B = randn(4,3,2)
To generate an array filled with a single constant value, use the repmat function. repmat replicates an array (in this case, a 1by1 array) through a vector of array dimensions.
B = repmat(5, [3 4 2]) B(:,:,1) = 5 5 5 5 5 5 5 5 5 5 5 5 B(:,:,2) = 5 5 5 5 5 5 5 5 5 5 5 5
Note Any dimension of an array can have size zero, making it a form of empty array. For example, 10by0by20 is a valid size for a multidimensional array. 
The cat function is a simple way to build multidimensional arrays; it concatenates a list of arrays along a specified dimension:
B = cat(dim, A1, A2...)
where A1, A2, and so on are the arrays to concatenate, and dim is the dimension along which to concatenate the arrays.
For example, to create a new array with cat:
B = cat(3, [2 8; 0 5], [1 3; 7 9]) B(:,:,1) = 2 8 0 5 B(:,:,2) = 1 3 7 9
The cat function accepts any combination of existing and new data. In addition, you can nest calls to cat. The lines below, for example, create a fourdimensional array.
A = cat(3, [9 2; 6 5], [7 1; 8 4]) B = cat(3, [3 5; 0 1], [5 6; 2 1]) D = cat(4, A, B, cat(3, [1 2; 3 4], [4 3;2 1]))
cat automatically adds subscripts of 1 between dimensions, if necessary. For example, to create a 2by2by1by2 array, enter
C = cat(4, [1 2; 4 5], [7 8; 3 2])
In the previous case, cat inserts as many singleton dimensions as needed to create a fourdimensional array whose last dimension is not a singleton dimension. If the dim argument had been 5, the previous statement would have produced a 2by2by1by1by2 array. This adds additional 1s to indexing expressions for the array. To access the value 8 in the fourdimensional case, use
You can use the following MATLAB functions to get information about multidimensional arrays you have created.
size — Returns the size of each array dimension.
size(C) ans = 2 2 1 2 rows columns dim3 dim4
ndims — Returns the number of dimensions in the array.
ndims(C) ans = 4
whos — Provides information on the format and storage of the array.
whos Name Size Bytes Class A 2x2x2 64 double array B 2x2x2 64 double array C 4D 64 double array D 4D 192 double array Grand total is 48 elements using 384 bytes
Many of the concepts that apply to twodimensional matrices extend to multidimensional arrays as well.
To access a single element of a multidimensional array, use integer subscripts. Each subscript indexes a dimension—the first indexes the row dimension, the second indexes the column dimension, the third indexes the first page dimension, and so on.
Consider a 10by5by3 array nddata of random integers:
nddata = fix(8 * randn(10,5,3));
To access element (3,2) on page 2 of nddata, for example, use nddata(3,2,2).
You can use vectors as array subscripts. In this case, each vector element must be a valid subscript, that is, within the bounds defined by the dimensions of the array. To access elements (2,1), (2,3), and (2,4) on page 3 of nddata, use
nddata(2,[1 3 4],3);
The MATLAB colon indexing extends to multidimensional arrays. For example, to access the entire third column on page 2 of nddata, use nddata(:,3,2).
The colon operator is also useful for accessing other subsets of data. For example, nddata(2:3,2:3,1) results in a 2by2 array, a subset of the data on page 1 of nddata. This matrix consists of the data in rows 2 and 3, columns 2 and 3, on the first page of the array.
The colon operator can appear as an array subscript on both sides of an assignment statement. For example, to create a 4by4 array of zeros:
C = zeros(4, 4)
Now assign a 2by2 subset of array nddata to the four elements in the center of C.
C(2:3,2:3) = nddata(2:3,1:2,2)
MATLAB linear indexing also extends to multidimensional arrays. In this case, MATLAB operates on a pagebypage basis to create the storage column, again appending elements columnwise. See Linear Indexing for an introduction to this topic.
For example, consider a 5by4by3by2 array C.
Again, a single subscript indexes directly into this column. For example, C(4) produces the result
ans = 0
If you specify two subscripts (i,j) indicating rowcolumn indices, MATLAB calculates the offset as described above. Two subscripts always access the first page of a multidimensional array, provided they are within the range of the original array dimensions.
If more than one subscript is present, all subscripts must conform to the original array dimensions. For example, C(6,2) is invalid because all pages of C have only five rows.
If you specify more than two subscripts, MATLAB extends its indexing scheme accordingly. For example, consider four subscripts (i,j,k,l) into a fourdimensional array with size [d1 d2 d3 d4]. MATLAB calculates the offset into the storage column by
(l1)(d3)(d2)(d1)+(k1)(d2)(d1)+(j1)(d1)+i
For example, if you index the array C using subscripts (3, 4, 2, 1), MATLAB returns the value 5 (index 38 in the storage column).
In general, the offset formula for an array with dimensions [d_{1} d_{2} d_{3} ... d_{n}] using any subscripts (s_{1} s_{2} s_{3} ... s_{n}) is
(s_{n}1)(d_{n1})(d_{n2})...(d_{1})+(s_{n1}1)(d_{n2})...(d_{1})+...+(s_{2}1)(d_{1})+s_{1 }
Because of this scheme, you can index an array using any number of subscripts. You can append any number of 1s to the subscript list because these terms become zero. For example,
C(3,2,1,1,1,1,1,1)
is equivalent to
C(3,2)
Some assignment statements, such as
A(:,:,2) = 1:10
are ambiguous because they do not provide enough information about the shape of the dimension to receive the data. In the case above, the statement tries to assign a onedimensional vector to a twodimensional destination. MATLAB produces an error for such cases. To resolve the ambiguity, be sure you provide enough information about the destination for the assigned data, and that both data and destination have the same shape. For example:
A(1,:,2) = 1:10;
Unless you change its shape or size, a MATLAB array retains the dimensions specified at its creation. You change array size by adding or deleting elements. You change array shape by respecifying the array's row, column, or page dimensions while retaining the same elements. The reshape function performs the latter operation. For multidimensional arrays, its form is
B = reshape(A,[s1 s2 s3 ...])
s1, s2, and so on represent the desired size for each dimension of the reshaped matrix. Note that a reshaped array must have the same number of elements as the original array (that is, the product of the dimension sizes is constant).
M  reshape(M, [6 5]) 



The reshape function operates in a columnwise manner. It creates the reshaped matrix by taking consecutive elements down each column of the original data construct.
C  reshape(C, [6 2]) 



Here are several new arrays from reshaping nddata:
B = reshape(nddata, [6 25]) C = reshape(nddata, [5 3 10]) D = reshape(nddata, [5 3 2 5])
MATLAB automatically removes trailing singleton dimensions (dimensions whose sizes are equal to 1) from all multidimensional arrays. For example, if you attempt to create an array of size 3by2by1by1, perhaps with the command rand(3,2,1,1), then MATLAB strips away the singleton dimensions at the end of the array and creates a 3by2 matrix. This is because every array technically has an infinite number of trailing singleton dimensions. A 3by2 array is the same as an array with size 3by2by1by1by1by...
For example, consider the following 2by2 matrix, A.
A = eye(2) A = 1 0 0 1
A is a 2by2 identity matrix.
Find the size of the fourth dimension of A.
size(A,4) ans = 1
Although A is a 2by2 matrix, the size of the fourth dimension in A is 1. In fact, the size of each dimension beyond the second is 1.
The first two dimensions of an array are never stripped away, since they are always relevant.
size(3) ans = 1 1
Even a scalar in MATLAB is a 1by1 matrix.
MATLAB creates singleton dimensions if you explicitly specify them when you create or reshape an array, or if you perform a calculation that results in an array dimension of one:
B = repmat(5, [2 3 1 4]); size(B) ans = 2 3 1 4
The squeeze function removes singleton dimensions from an array:
C = squeeze(B); size(C) ans = 2 3 4
The squeeze function does not affect twodimensional arrays; row vectors remain rows.
The permute function reorders the dimensions of an array:
B = permute(A, dims);
dims is a vector specifying the new order for the dimensions of A, where 1 corresponds to the first dimension (rows), 2 corresponds to the second dimension (columns), 3 corresponds to pages, and so on.
For a more detailed look at the permute function, consider a fourdimensional array A of size 5by4by3by2. Rearrange the dimensions, placing the column dimension first, followed by the second page dimension, the first page dimension, then the row dimension. The result is a 4by2by3by5 array.
You can think of permute's operation as an extension of the transpose function, which switches the row and column dimensions of a matrix. For permute, the order of the input dimension list determines the reordering of the subscripts. In the example above, element (4,2,1,2) of A becomes element (2,2,1,4) of B, element (5,4,3,2) of A becomes element (4,2,3,5) of B, and so on.
The ipermute function is the inverse of permute. Given an input array A and a vector of dimensions v, ipermute produces an array B such that permute(B,v) returns A.
For example, these statements create an array E that is equal to the input array C:
D = ipermute(C, [1 4 2 3]); E = permute(D, [1 4 2 3])
You can obtain the original array after permuting it by calling ipermute with the same vector of dimensions.
Many of the MATLAB computational and mathematical functions accept multidimensional arrays as arguments. These functions operate on specific dimensions of multidimensional arrays; that is, they operate on individual elements, on vectors, or on matrices.
Functions that operate on vectors, like sum, mean, and so on, by default typically work on the first nonsingleton dimension of a multidimensional array. Most of these functions optionally let you specify a particular dimension on which to operate. There are exceptions, however. For example, the cross function, which finds the cross product of two vectors, works on the first nonsingleton dimension having length 3.
Note In many cases, these functions have other restrictions on the input arguments — for example, some functions that accept multiple arrays require that the arrays be the same size. Refer to the online help for details on function arguments. 
MATLAB functions that operate elementbyelement on twodimensional arrays, like the trigonometric and exponential functions in the elfun directory, work in exactly the same way for multidimensional cases. For example, the sin function returns an array the same size as the function's input argument. Each element of the output array is the sine of the corresponding element of the input array.
Similarly, the arithmetic, logical, and relational operators all work with corresponding elements of multidimensional arrays that are the same size in every dimension. If one operand is a scalar and one an array, the operator applies the scalar to each element of the array.
Functions that operate on planes or matrices, such as the linear algebra and matrix functions in the matfun directory, do not accept multidimensional arrays as arguments. That is, you cannot use the functions in the matfun directory, or the array operators *, ^, \, or /, with multidimensional arguments. Supplying multidimensional arguments or operands in these cases results in an error.
You can use indexing to apply a matrix function or operator to matrices within a multidimensional array. For example, create a threedimensional array A:
A = cat(3, [1 2 3; 9 8 7; 4 6 5], [0 3 2; 8 8 4; 5 3 5], ... [6 4 7; 6 8 5; 5 4 3]);
Applying the eig function to the entire multidimensional array results in an error:
eig(A) ??? Undefined function or method 'eig' for input arguments of type 'double' and attributes 'full 3d real'.
You can, however, apply eig to planes within the array. For example, use colon notation to index just one page (in this case, the second) of the array:
eig(A(:,:,2)) ans = 12.9129 2.6260 2.7131
Note In the first case, subscripts are not colons; you must use squeeze to avoid an error. For example, eig(A(2,:,:)) results in an error because the size of the input is [1 3 3]. The expression eig(squeeze(A(2,:,:))), however, passes a valid twodimensional matrix to eig. 
You can use multidimensional arrays to represent data in two ways:
As planes or pages of twodimensional data. You can then treat these pages as matrices.
As multivariate or multidimensional data. For example, you might have a fourdimensional array where each element corresponds to either a temperature or air pressure measurement taken at one of a set of equally spaced points in a room.
For example, consider an RGB image. For a single image, a multidimensional array is probably the easiest way to store and access data.
To access an entire plane of the image, use
redPlane = RGB(:,:,1);
To access a subimage, use
subimage = RGB(20:40,50:85,:);
The RGB image is a good example of data that needs to be accessed in planes for operations like display or filtering. In other instances, however, the data itself might be multidimensional. For example, consider a set of temperature measurements taken at equally spaced points in a room. Here the location of each value is an integral part of the data set—the physical placement in threespace of each element is an aspect of the information. Such data also lends itself to representation as a multidimensional array.
Now to find the average of all the measurements, use
mean(mean(mean(TEMP)));
To obtain a vector of the "middle" values (element (2,2)) in the room on each page, use
B = TEMP(2,2,:);
Like numeric arrays, the framework for multidimensional cell arrays in MATLAB is an extension of the twodimensional cell array model. You can use the cat function to build multidimensional cell arrays, just as you use it for numeric arrays.
For example, create a simple threedimensional cell array C:
A{1,1} = [1 2;4 5]; A{1,2} = 'Name'; A{2,1} = 24i; A{2,2} = 7; B{1,1} = 'Name2'; B{1,2} = 3; B{2,1} = 0:1:3; B{2,2} = [4 5]'; C = cat(3, A, B);
The subscripts for the cells of C look like
Multidimensional structure arrays are extensions of rectangular structure arrays. Like other types of multidimensional arrays, you can build them using direct assignment or the cat function:
patient(1,1,1).name = 'John Doe'; patient(1,1,1).billing = 127.00; patient(1,1,1).test = [79 75 73; 180 178 177.5; 220 210 205]; patient(1,2,1).name = 'Ann Lane'; patient(1,2,1).billing = 28.50; patient(1,2,1).test = [68 70 68; 118 118 119; 172 170 169]; patient(1,1,2).name = 'Al Smith'; patient(1,1,2).billing = 504.70; patient(1,1,2).test = [80 80 80; 153 153 154; 181 190 182]; patient(1,2,2).name = 'Dora Jones'; patient(1,2,2).billing = 1173.90; patient(1,2,2).test = [73 73 75; 103 103 102; 201 198 200];
To apply functions to multidimensional structure arrays, operate on fields and field elements using indexing. For example, find the sum of the columns of the test array in patient(1,1,2):
sum((patient(1,1,2).test));
Similarly, add all the billing fields in the patient array:
total = sum([patient.billing]);