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### 16.2 Rearranging Matrices

: B = fliplr (A)

Flip array left to right.

Return a copy of A with the order of the columns reversed. In other words, A is flipped left-to-right about a vertical axis. For example:

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

: B = flipud (A)

Flip array upside down.

Return a copy of A with the order of the rows reversed. In other words, A is flipped upside-down about a horizontal axis. For example:

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

: B = flip (A)
: B = flip (A, dim)

Return a copy of array A flipped across dimension dim.

If dim is unspecified it defaults to the first non-singleton dimension.

Examples:

```## row vector
flip ([1  2  3  4])
⇒  4  3  2  1

## column vector
flip ([1; 2; 3; 4])
⇒  4
3
2
1

## 2-D matrix along dimension 1
flip ([1 2; 3 4])
⇒  3  4
1  2

## 2-D matrix along dimension 2
flip ([1 2; 3 4], 2)
⇒  2  1
4  3
```

: B = rot90 (A)
: B = rot90 (A, k)

Rotate array by 90 degree increments.

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)
```

The rotation is always performed on the plane of the first two dimensions, i.e., rows and columns. To perform a rotation on any other plane, use `rotdim`.

: B = rotdim (A)
: B = rotdim (A, n)
: B = rotdim (A, n, plane)

Return a copy of A with the elements rotated counterclockwise in 90-degree increments.

The second argument n is optional, and specifies how many 90-degree rotations are to be applied (the default value is 1). Negative values of n rotate the matrix in a clockwise direction.

The third argument is also optional and defines the plane of the rotation. If present, plane is a two element vector containing two different valid dimensions of the matrix. When plane is not given the first two non-singleton dimensions are used.

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])
```

: A = 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)
⇒ 1 1 0 0
1 1 0 0
```

Alternatively, we can concatenate A and B along the second dimension in 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(:,:,1,1) =

1 1
1 1

ans(:,:,1,2) =

0 0
0 0
```

: A = horzcat (array1, array2, …, arrayN)

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

Arrays may also be concatenated horizontally using the syntax for creating new matrices. For example:

```A = [ array1, array2, … ]
```

This syntax is slightly more efficient because the Octave parser can concatenate the arrays without the overhead of a function call.

: A = vertcat (array1, array2, …, arrayN)

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

Arrays may also be concatenated vertically using the syntax for creating new matrices. For example:

```A = [ array1; array2; … ]
```

This syntax is slightly more efficient because the Octave parser can concatenate the arrays without the overhead of a function call.

: B = 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 only once). The Nth dimension of A gets remapped to dimension `PERM(N)`. For example:

```x = zeros ([2, 3, 5, 7]);
size (x)
⇒  2   3   5   7

size (permute (x, [2, 1, 3, 4]))
⇒  3   2   5   7

size (permute (x, [1, 3, 4, 2]))
⇒  2   5   7   3

## The identity permutation
size (permute (x, [1, 2, 3, 4]))
⇒  2   3   5   7
```

: A = ipermute (B, iperm)

The inverse of the `permute` function.

The expression

```ipermute (permute (A, perm), perm)
```

returns the original array A.

: B = reshape (A, m, n, …)
: B = reshape (A, [m n …])
: B = reshape (A, …, [], …)
: B = reshape (A, size)

Return a matrix with the specified dimensions (m, n, …) whose elements are taken from the matrix A.

The elements of the matrix are accessed in column-major order (like Fortran arrays are stored).

The following code demonstrates reshaping a 1x4 row vector into a 2x2 square matrix.

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

Note that the total number of elements in the original matrix (`prod (size (A))`) must match the total number of elements in the new matrix (`prod ([m n …])`).

A single dimension of the return matrix may be left unspecified and Octave will determine its size automatically. An empty matrix ([]) is used to flag the unspecified dimension.

: B = resize (A, m)
: B = resize (A, m, n, …)
: B = resize (A, [m n …])

Resize A cutting off elements as necessary.

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

In other words, the statement

```B = resize (A, dv)
```

is equivalent to the following code:

```B = zeros (dv, class (A));
sz = min (dv, size (A));
for i = 1:length (sz)
idx{i} = 1:sz(i);
endfor
B(idx{:}) = A(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, n, … are all scalars, then the dimensions of the result are m-by-n-by-…. If given a vector as input, then the dimensions of the result are given by the elements of that vector.

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.

: y = circshift (x, n)
: y = circshift (x, n, dim)

Circularly shift 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 of x are shifted. If an element of n is zero, then the corresponding dimension of x will not be shifted. If n is a scalar and no dim is specified then the shift is applied to the first non-singular dimension.

If a scalar dim is given then operate along the specified dimension. In this case n must be a scalar as well.

Examples:

```x = [1, 2, 3;
4, 5, 6;
7, 8, 9];
## positive shift on rows (1st non-singular dim)
circshift (x, 1)
⇒
7   8   9
1   2   3
4   5   6
## negative shift on rows (1st non-singular dim)
circshift (x, -2)
⇒
7   8   9
1   2   3
4   5   6
## no shift of rows, shift columns by 1 (2nd dimension)
circshift (x, [0,1])
⇒
3   1   2
6   4   5
9   7   8
## shift columns (2nd dimension)
circshift (x, 1, 2)
⇒
3   1   2
6   4   5
9   7   8
```

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

Shift the dimensions 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
```

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

Return a copy of x with the elements arranged in increasing order.

For matrices, `sort` orders the elements within columns

For example:

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

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"`.

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
```

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

Sorting of complex entries is done first by magnitude (`abs (z)`) and for any ties by phase angle (`angle (z)`). For example:

```sort ([1+i; 1; 1-i])
⇒ 1 + 0i
1 - 1i
1 + 1i
```

NaN values are treated as being greater than any other value and are sorted to the end of the list.

The `sort` function may also be used to sort strings and cell arrays of strings, in which case ASCII dictionary order (uppercase ’A’ precedes lowercase ’a’) of the strings is used.

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

: [s, i] = sortrows (A)
: [s, i] = sortrows (A, c)

Sort the rows of the matrix A according to the order of the columns specified in c.

By default (c omitted, or a particular column unspecified in c) an ascending sort order is used. However, if elements of c are negative then the corresponding column is sorted in descending order. If the elements of A are strings then a lexicographical sort is used.

Example: sort by column 2 in descending order, then 3 in ascending order

```x = [ 7, 1, 4;
8, 3, 5;
9, 3, 6 ];
sortrows (x, [-2, 3])
⇒ 8  3  5
9  3  6
7  1  4
```

: tf = issorted (A)
: tf = issorted (A, mode)
: tf = issorted (A, "rows", mode)

Return true if the vector A is sorted according to mode, which may be either `"ascend"`, `"descend"`, or `"either"`.

By default, mode is `"ascend"`. NaNs are treated in the same manner as `sort`.

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

This function does not support sparse matrices.

: nel = nth_element (x, n)
: nel = nth_element (x, n, dim)

Select the n-th smallest element of a vector, using the ordering defined by `sort`.

The result is equivalent to `sort(x)(n)`.

n can also be a contiguous range, either ascending `l:u` or descending `u:-1:l`, in which case a range of elements is returned.

If x is an array, `nth_element` operates along the dimension defined by dim, or the first non-singleton dimension if dim is not given.

Programming Note: nth_element encapsulates the C++ standard library algorithms nth_element and partial_sort. On average, the complexity of the operation is O(M*log(K)), where `M = size (x, dim)` and `K = length (n)`. This function is intended for cases where the ratio K/M is small; otherwise, it may be better to use `sort`.

: A_LO = tril (A)
: A_LO = tril (A, k)
: A_LO = tril (A, k, pack)

Return a new matrix formed by extracting the lower triangular part of the matrix A, and setting all other elements to zero.

The optional second argument specifies how many diagonals above or below the main diagonal should also be set to zero. The default value of k is zero which includes the main diagonal as part of the result. If the value of k is a nonzero integer then the selection of elements starts at an offset of k diagonals above the main diagonal for positive k or below the main diagonal for negative k. The absolute value of k may not be greater than the number of subdiagonals or superdiagonals.

Example 1 : exclude main diagonal

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

Example 2 : include first superdiagonal

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

If the optional third argument `"pack"` is given then the extracted elements are not inserted into a matrix, but instead stacked column-wise one above another, and returned as a column vector.

: A_UP = triu (A)
: A_UP = triu (A, k)
: A_UP = triu (A, k, pack)

Return a new matrix formed by extracting the upper triangular part of the matrix A, and setting all other elements to zero.

The optional second argument specifies how many diagonals above or below the main diagonal should also be set to zero. The default value of k is zero which includes the main diagonal as part of the result. If the value of k is a nonzero integer then the selection of elements starts at an offset of k diagonals above the main diagonal for positive k or below the main diagonal for negative k. The absolute value of k may not be greater than the number of subdiagonals or superdiagonals.

Example 1 : exclude main diagonal

```triu (ones (3), 1)
⇒  0  1  1
0  0  1
0  0  0
```

Example 2 : include first subdiagonal

```triu (ones (3), -1)
⇒  1  1  1
1  1  1
0  1  1
```

If the optional third argument `"pack"` is given then the extracted elements are not inserted into a matrix, but instead stacked column-wise one above another, and returned as a column vector.

: v = vec (x)
: 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)`.

: v = vech (x)

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

This has uses in matrix calculus where the underlying matrix is symmetric and it would be pointless to keep values above the main diagonal.

: B = prepad (A, l)
: B = prepad (A, l, c)
: B = prepad (A, l, c, dim)

Prepend the scalar value c to the vector A until it is of length l. If c is not given, a value of 0 is used.

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

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

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

If dim is larger than the dimensions of A, the result will have dim dimensions.

: B = postpad (A, l)
: B = postpad (A, l, c)
: B = postpad (A, l, c, dim)

Append the scalar value c to the vector A until it is of length l. If c is not given, a value of 0 is used.

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

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

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

If dim is larger than the dimensions of A, the result will have dim dimensions.

: M = diag (v)
: M = diag (v, k)
: M = diag (v, m, n)
: v = diag (M)
: v = diag (M, 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 superdiagonal. If it is negative, it is placed on the -k-th subdiagonal. 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
```

The 3-input form returns a diagonal matrix with vector v on the main diagonal and the resulting matrix being of size m rows x n columns.

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

: M = blkdiag (A, B, C, …)

Build a block diagonal matrix from A, B, C, …

All arguments must be numeric and either two-dimensional matrices or scalars. If any argument is of type sparse, the output will also be sparse.