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# Matrix Manipulation

There are a number of functions available for checking to see if the elements of a matrix meet some condition, and for rearranging the elements of a matrix. For example, Octave can easily tell you if all the elements of a matrix are finite, or are less than some specified value. Octave can also rotate the elements, extract the upper- or lower-triangular parts, or sort the columns of a matrix.

## Finding Elements and Checking Conditions

The functions `any` and `all` are useful for determining whether any or all of the elements of a matrix satisfy some condition. The `find` function is also useful in determining which elements of a matrix meet a specified condition.

Built-in Function: any (x)
For a vector argument, return 1 if any element of the vector is nonzero.

For a matrix argument, return a row vector of ones and zeros with each element indicating whether any of the elements of the corresponding column of the matrix are nonzero. For example,

```any (eye (2, 4))

=> [ 1, 1, 0, 0 ]
```

To see if any of the elements of a matrix are nonzero, you can use a statement like

```any (any (a))
```

Built-in Function: all (x)
The function `all` behaves like the function `any`, except that it returns true only if all the elements of a vector, or all the elements in a column of a matrix, are nonzero.

Since the comparison operators (see section Comparison Operators) return matrices of ones and zeros, it is easy to test a matrix for many things, not just whether the elements are nonzero. For example,

```all (all (rand (5) < 0.9))

=> 0
```

tests a random 5 by 5 matrix to see if all of it's elements are less than 0.9.

Note that in conditional contexts (like the test clause of `if` and `while` statements) Octave treats the test as if you had typed `all (all (condition))`.

Function File: [errorcode, y_1, ...] = common_size (x_1, ...)
Determine if all input arguments are either scalar or of common size. If so, errorcode is zero, and y_i is a matrix of the common size with all entries equal to x_i if this is a scalar or x_i otherwise. If the inputs cannot be brought to a common size, errorcode is 1, and y_i is x_i. For example,

```[errorcode, a, b] = common_size ([1 2; 3 4], 5)
=> errorcode = 0
=> a = [1 2, 3 4]
=> b = [5 5; 5 5]
```

This is useful for implementing functions where arguments can either be scalars or of common size.

Function File: diff (x, k)
If x is a vector of length n, `diff (x)` is the vector of first differences

If x is a matrix, `diff (x)` is the matrix of column differences.

The second argument is optional. If supplied, ```diff (x, k)```, where k is a nonnegative integer, returns the k-th differences.

Mapping Function: isinf (x)
Return 1 for elements of x that are infinite and zero otherwise. For example,

```isinf ([13, Inf, NaN])

=> [ 0, 1, 0 ]
```

Mapping Function: isnan (x)
Return 1 for elements of x that are NaN values and zero otherwise. For example,

```isnan ([13, Inf, NaN])

=> [ 0, 0, 1 ]
```

Mapping Function: finite (x)
Return 1 for elements of x that are NaN values and zero otherwise. For example,

```finite ([13, Inf, NaN])

=> [ 1, 0, 0 ]
```

Loadable Function: find (x)
The function `find` returns a vector of indices of nonzero elements of a matrix. To obtain a single index for each matrix element, Octave pretends that the columns of a matrix form one long vector (like Fortran arrays are stored). For example,

```find (eye (2))

=> [ 1; 4 ]
```

If two outputs are requested, `find` returns the row and column indices of nonzero elements of a matrix. For example,

```[i, j] = find (2 * eye (2))

=> i = [ 1; 2 ]

=> j = [ 1; 2 ]
```

If three outputs are requested, `find` also returns a vector containing the the nonzero values. For example,

```[i, j, v] = find (3 * eye (2))

=> i = [ 1; 2 ]

=> j = [ 1; 2 ]

=> v = [ 3; 3 ]
```

## 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
```

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
```

Function File: rot90 (x, n)
Returns 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). Negative values of n 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)
```

Function File: reshape (a, m, n)
Return a matrix with m rows and n columns whose elements are taken from the matrix a. To decide how to order the elements, Octave pretends that the elements of a matrix are stored in column-major order (like Fortran arrays are stored).

For example,

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

=>  1  3
2  4
```

If the variable `do_fortran_indexing` is nonzero, the `reshape` function is equivalent to

```retval = zeros (m, n);
retval (:) = a;
```

but it is somewhat less cryptic to use `reshape` instead of the colon operator. Note that the total number of elements in the original matrix must match the total number of elements in the new matrix.

Function File: shift (x, b)
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.

Loadable Function: [s, i] = sort (x)
Returns a copy of x with the elements 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
```

Since the `sort` function does not allow sort keys to be specified, so it can't be used to order the rows of a matrix according to the values of the elements in various columns(6) 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, k)
Function File: triu (a, k)
Return a new matrix form by extracting extract 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 matrix.

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- 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
```

Function File: vec (x)
For a matrix x, returns the vector obtained by stacking the columns of x one above the other.

See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

Function File: vech (x)
For a square matrix x, returns the vector obtained from x by eliminating all supradiagonal elements and stacking the result one column above the other.

See Magnus and Neudecker (1988), Matrix differential calculus with applications in statistics and econometrics.

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