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### 16.2 Rearranging Matrices

— Function File: fliplr (x)

> Return a copy of x with the order of the columns reversed. For example:

>               fliplr ([1, 2; 3, 4])
>                    ⇒  2  1
>                        4  3

> Note that `fliplr` only work with 2-D arrays. To flip N-d arrays use `flipdim` instead.

> See also: flipud, flipdim, rot90, rotdim.

— Function File: flipud (x)

> Return a copy of x with the order of the rows reversed. For example:

>               flipud ([1, 2; 3, 4])
>                    ⇒  3  4
>                        1  2

> Due to the difficulty of defining which axis about which to flip the matrix `flipud` only work with 2-d arrays. To flip N-d arrays use `flipdim` instead.

> See also: fliplr, flipdim, rot90, rotdim.

— Function File: flipdim (x)
— Function File: flipdim (x, dim)

> Return a copy of x flipped about the dimension dim. dim defaults to the first non-singleton dimension. For example:

>               flipdim ([1, 2; 3, 4], 2)
>                    ⇒  2  1
>                        4  3

> See also: fliplr, flipud, rot90, rotdim.

— Function File: rot90 (A, k)

> Return a copy of A with the elements rotated counterclockwise in 90-degree increments. The second argument is optional, and specifies how many 90-degree rotations are to be applied (the default value is 1). Negative values of k rotate the matrix in a clockwise direction. For example,

>               rot90 ([1, 2; 3, 4], -1)
>                    ⇒  3  1
>                        4  2

> rotates the given matrix clockwise by 90 degrees. The following are all equivalent statements:

>               rot90 ([1, 2; 3, 4], -1)
>               rot90 ([1, 2; 3, 4], 3)
>               rot90 ([1, 2; 3, 4], 7)

> Due to the difficulty of defining an axis about which to rotate the matrix `rot90` only work with 2-D arrays. To rotate N-d arrays use `rotdim` instead.

> See also: rotdim, flipud, fliplr, flipdim.

— Function File: rotdim (x, n, plane)

> Return a copy of x with the elements rotated counterclockwise in 90-degree increments. The second argument is optional, and specifies how many 90-degree rotations are to be applied (the default value is 1). The third argument is also optional and defines the plane of the rotation. As such plane is a two element vector containing two different valid dimensions of the matrix. If plane is not given Then the first two non-singleton dimensions are used.

> Negative values of n rotate the matrix in a clockwise direction. For example,

>               rotdim ([1, 2; 3, 4], -1, [1, 2])
>                    ⇒  3  1
>                        4  2

> rotates the given matrix clockwise by 90 degrees. The following are all equivalent statements:

>               rotdim ([1, 2; 3, 4], -1, [1, 2])
>               rotdim ([1, 2; 3, 4], 3, [1, 2])
>               rotdim ([1, 2; 3, 4], 7, [1, 2])

> See also: rot90, flipud, fliplr, flipdim.

— Built-in Function: cat (dim, array1, array2, ..., arrayN)

> Return the concatenation of N-d array objects, array1, array2, ..., arrayN along dimension dim.

>               A = ones (2, 2);
>               B = zeros (2, 2);
>               cat (2, A, B)
>               ⇒ ans =

>                    1 1 0 0
>                    1 1 0 0

> Alternatively, we can concatenate A and B along the second dimension the following way:

>               [A, B].

> dim can be larger than the dimensions of the N-d array objects and the result will thus have dim dimensions as the following example shows:

>               cat (4, ones(2, 2), zeros (2, 2))
>               ⇒ ans =

>                  ans(:,:,1,1) =

>                    1 1
>                    1 1

>                  ans(:,:,1,2) =
>                    0 0
>                    0 0

> See also: horzcat, vertcat.

— Built-in Function: horzcat (array1, array2, ..., arrayN)

> Return the horizontal concatenation of N-d array objects, array1, array2, ..., arrayN along dimension 2.

> See also: cat, vertcat.

— Built-in Function: vertcat (array1, array2, ..., arrayN)

> Return the vertical concatenation of N-d array objects, array1, array2, ..., arrayN along dimension 1.

> See also: cat, horzcat.

— Built-in Function: permute (a, perm)

> Return the generalized transpose for an N-d array object a. The permutation vector perm must contain the elements `1:ndims(a)` (in any order, but each element must appear just once).

> See also: ipermute.

— Built-in Function: ipermute (a, iperm)

> The inverse of the `permute` function. The expression

>               ipermute (permute (a, perm), perm)

> returns the original array a.

> See also: permute.

— Built-in Function: reshape (a, m, n, ...)
— Built-in Function: reshape (a, size)

> Return a matrix with the given dimensions whose elements are taken from the matrix a. The elements of the matrix are accessed in column-major order (like Fortran arrays are stored).

> For example:

>               reshape ([1, 2, 3, 4], 2, 2)
>                    ⇒  1  3
>                        2  4

> Note that the total number of elements in the original matrix must match the total number of elements in the new matrix.

> A single dimension of the return matrix can be unknown and is flagged by an empty argument.

— Built-in Function: resize (x, m)
— Built-in Function: resize (x, m, n)
— Built-in Function: resize (x, m, n, ...)

> Resize x cutting off elements as necessary.

> In the result, element with certain indices is equal to the corresponding element of x if the indices are within the bounds of x; otherwise, the element is set to zero.

> In other words, the statement

>                 y = resize (x, dv);

> is equivalent to the following code:

>                 y = zeros (dv, class (x));
>                 sz = min (dv, size (x));
>                 for i = 1:length (sz), idx{i} = 1:sz(i); endfor
>                 y(idx{:}) = x(idx{:});

> but is performed more efficiently.

> If only m is supplied and it is a scalar, the dimension of the result is m-by-m. If m is a vector, then the dimensions of the result are given by the elements of m. If both m and n are scalars, then the dimensions of the result are m-by-n.

> An object can be resized to more dimensions than it has; in such case the missing dimensions are assumed to be 1. Resizing an object to fewer dimensions is not possible.

> See also: reshape, postpad.

— Function File: y = circshift (x, n)

> Circularly shifts the values of the array x. n must be a vector of integers no longer than the number of dimensions in x. The values of n can be either positive or negative, which determines the direction in which the values or x are shifted. If an element of n is zero, then the corresponding dimension of x will not be shifted. For example:

>               x = [1, 2, 3; 4, 5, 6; 7, 8, 9];
>               circshift (x, 1)
>               ⇒  7, 8, 9
>                   1, 2, 3
>                   4, 5, 6
>               circshift (x, -2)
>               ⇒  7, 8, 9
>                   1, 2, 3
>                   4, 5, 6
>               circshift (x, [0,1])
>               ⇒  3, 1, 2
>                   6, 4, 5
>                   9, 7, 8

> See also: permute, ipermute, shiftdim.

— Function File: y = shiftdim (x, n)
— Function File: [y, ns] = shiftdim (x)

> Shifts the dimension of x by n, where n must be an integer scalar. When n is positive, the dimensions of x are shifted to the left, with the leading dimensions circulated to the end. If n is negative, then the dimensions of x are shifted to the right, with n leading singleton dimensions added.

> Called with a single argument, `shiftdim`, removes the leading singleton dimensions, returning the number of dimensions removed in the second output argument ns.

> For example:

>               x = ones (1, 2, 3);
>               size (shiftdim (x, -1))
>                    ⇒ [1, 1, 2, 3]
>               size (shiftdim (x, 1))
>                    ⇒ [2, 3]
>               [b, ns] = shiftdim (x);
>                    ⇒ b =  [1, 1, 1; 1, 1, 1]
>                    ⇒ ns = 1

> See also: reshape, permute, ipermute, circshift, squeeze.

— Function File: shift (x, b)
— Function File: shift (x, b, dim)

> If x is a vector, perform a circular shift of length b of the elements of x.

> If x is a matrix, do the same for each column of x. If the optional dim argument is given, operate along this dimension.

— Loadable Function: [s, i] = sort (x)
— Loadable Function: [s, i] = sort (x, dim)
— Loadable Function: [s, i] = sort (x, mode)
— Loadable Function: [s, i] = sort (x, dim, mode)

> Return a copy of x with the elements arranged in increasing order. For matrices, `sort` orders the elements in each column.

> For example:

>               sort ([1, 2; 2, 3; 3, 1])
>                    ⇒  1  1
>                        2  2
>                        3  3

> The `sort` function may also be used to produce a matrix containing the original row indices of the elements in the sorted matrix. For example:

>               [s, i] = sort ([1, 2; 2, 3; 3, 1])
>                    ⇒ s = 1  1
>                           2  2
>                           3  3
>                    ⇒ i = 1  3
>                           2  1
>                           3  2

> If the optional argument dim is given, then the matrix is sorted along the dimension defined by dim. The optional argument `mode` defines the order in which the values will be sorted. Valid values of `mode` are `ascend' or `descend'.

> For equal elements, the indices are such that the equal elements are listed in the order that appeared in the original list.

> The `sort` function may also be used to sort strings and cell arrays of strings, in which case the dictionary order of the strings is used.

> The algorithm used in `sort` is optimized for the sorting of partially ordered lists.

— Function File: sortrows (A, c)

> Sort the rows of the matrix A according to the order of the columns specified in c. If c is omitted, a lexicographical sort is used. By default ascending order is used however if elements of c are negative then the corresponding column is sorted in descending order.

— Built-in Function: issorted (a, mode)
— Built-in Function: issorted (a, `"rows"`, mode)

> Return true if the array is sorted according to mode, which may be either "ascending", "descending", or "either". By default, mode is "ascending". NaNs are treated in the same manner as `sort`.

> If the optional argument "rows" is supplied, check whether the array is sorted by rows as output by the function `sortrows` (with no options).

> This function does not support sparse matrices.

> See also: sort, sortrows.

Since the `sort` function does not allow sort keys to be specified, it can't be used to order the rows of a matrix according to the values of the elements in various columns1 in a single call. Using the second output, however, it is possible to sort all rows based on the values in a given column. Here's an example that sorts the rows of a matrix based on the values in the second column.

         a = [1, 2; 2, 3; 3, 1];
         [s, i] = sort (a (:, 2));
         a (i, :)
              ⇒  3  1
                  1  2
                  2  3

— Function File: tril (A)
— Function File: tril (A, k)
— Function File: tril (A, k, pack)
— Function File: triu (A)
— Function File: triu (A, k)
— Function File: triu (A, k, pack)

> Return a new matrix formed by extracting the lower (`tril`) or upper (`triu`) triangular part of the matrix A, and setting all other elements to zero. The second argument is optional, and specifies how many diagonals above or below the main diagonal should also be set to zero.

> The default value of k is zero, so that `triu` and `tril` normally include the main diagonal as part of the result.

> If the value of k is negative, additional elements above (for `tril`) or below (for `triu`) the main diagonal are also selected.

> The absolute value of k must not be greater than the number of sub-diagonals or super-diagonals.

> For example:

>               tril (ones (3), -1)
>                    ⇒  0  0  0
>                        1  0  0
>                        1  1  0

> and

>               tril (ones (3), 1)
>                    ⇒  1  1  0
>                        1  1  1
>                        1  1  1

> If the option "pack" is given as third argument, the extracted elements are not inserted into a matrix, but rather stacked column-wise one above other.

> See also: triu, diag.

— Built-in Function: v = vec (x)
— Built-in Function: v = vec (x, dim)

> Return the vector obtained by stacking the columns of the matrix x one above the other. Without dim this is equivalent to x`(:)`. If dim is supplied, the dimensions of v are set to dim with all elements along the last dimension. This is equivalent to `shiftdim (`x`(:), 1-`dim`)`.

— Function File: vech (x)

> Return the vector obtained by eliminating all supradiagonal elements of the square matrix x and stacking the result one column above the other.

— Function File: prepad (x, l)
— Function File: prepad (x, l, c)
— Function File: prepad (x, l, c, dim)

> Prepend the scalar value c to the vector x until it is of length l. If the third argument is not supplied, a value of 0 is used.

> If `length (`x`) >` l, elements from the beginning of x are removed until a vector of length l is obtained.

> If x is a matrix, elements are prepended or removed from each row.

> If the optional argument dim is given, operate along this dimension.

> See also: postpad, cat, resize.

— Function File: postpad (x, l)
— Function File: postpad (x, l, c)
— Function File: postpad (x, l, c, dim)

> Append the scalar value c to the vector x until it is of length l. If the third argument is not supplied, a value of 0 is used.

> If `length (`x`) >` l, elements from the end of x are removed until a vector of length l is obtained.

> If x is a matrix, elements are appended or removed from each row.

> If the optional argument dim is given, operate along this dimension.

> See also: prepad, cat, resize.

— Built-in Function: diag (v, k)

> Return a diagonal matrix with vector v on diagonal k. The second argument is optional. If it is positive, the vector is placed on the k-th super-diagonal. If it is negative, it is placed on the -k-th sub-diagonal. The default value of k is 0, and the vector is placed on the main diagonal. For example:

>               diag ([1, 2, 3], 1)
>                    ⇒  0  1  0  0
>                        0  0  2  0
>                        0  0  0  3
>                        0  0  0  0

> Given a matrix argument, instead of a vector, `diag` extracts the k-th diagonal of the matrix.

— Function File: blkdiag (A, B, C, ...)

> Build a block diagonal matrix from A, B, C, .... All the arguments must be numeric and are two-dimensional matrices or scalars.

> See also: diag, horzcat, vertcat.

#### Footnotes

[1] For example, to first sort based on the values in column 1, and then, for any values that are repeated in column 1, sort based on the values found in column 2, etc.