# 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.

Given a vector, the function `any` returns 1 if any element of the vector is nonzero.

For a matrix argument, `any` returns 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,

```octave:13> any (eye (2, 4))
ans =

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))
```

For a matrix argument, `any` returns 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.

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,

```octave:13> all (all (rand (5) < 0.9))
ans = 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))`.

The functions `isinf`, `finite`, and `isnan` return 1 if their arguments are infinite, finite, or not a number, respectively, and return 0 otherwise. For matrix values, they all work on an element by element basis. For example, evaluating the expression

```isinf ([1, 2; Inf, 4])
```

produces the matrix

```ans =

0  0
1  0
```

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,

```octave:13> find (eye (2))
ans =

1
4
```

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

```octave:13> [i, j] = find (eye (2))
i =

1
2

j =

1
2
```

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

## Rearranging Matrices

The function `fliplr` reverses the order of the columns in a matrix, and `flipud` reverses the order of the rows. For example,

```octave:13> fliplr ([1, 2; 3, 4])
ans =

2  1
4  3

octave:13> flipud ([1, 2; 3, 4])
ans =

3  4
1  2
```

The function `rot90 (a, n)` rotates a matrix 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)
ans =

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 function `reshape (a, m, n)` returns 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,

```octave:13> reshape ([1, 2, 3, 4], 2, 2)
ans =

1  3
2  4
```

If the variable `do_fortran_indexing` is `"true"`, 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.

The function `sort' can be used to arrange the elements of a vector in increasing order. For matrices, `sort` orders the elements in each column.

For example,

```octave:13> sort (rand (4))
ans =

0.065359  0.039391  0.376076  0.384298
0.111486  0.140872  0.418035  0.824459
0.269991  0.274446  0.421374  0.938918
0.580030  0.975784  0.562145  0.954964
```

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 =

0.051724  0.485904  0.253614  0.348008
0.391608  0.526686  0.536952  0.600317
0.733534  0.545522  0.691719  0.636974
0.986353  0.976130  0.868598  0.713884

i =

2  4  2  3
4  1  3  4
1  2  4  1
3  3  1  2
```

These values may be used to recover the original matrix from the sorted version. For example,

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(5) 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 third column.

```octave:13> a = rand (4)
a =

0.080606  0.453558  0.835597  0.437013
0.277233  0.625541  0.447317  0.952203
0.569785  0.528797  0.319433  0.747698
0.385467  0.124427  0.883673  0.226632

octave:14> [s, i] = sort (a (:, 3));
octave:15> a (i, :)
ans =

0.569785  0.528797  0.319433  0.747698
0.277233  0.625541  0.447317  0.952203
0.080606  0.453558  0.835597  0.437013
0.385467  0.124427  0.883673  0.226632
```

The functions `triu (a, k)` and ```tril (a, k)``` extract the upper or lower triangular part of the matrix a, and set 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,

```octave:13> tril (rand (4), 1)
ans =

0.00000  0.00000  0.00000  0.00000
0.09012  0.00000  0.00000  0.00000
0.01215  0.34768  0.00000  0.00000
0.00302  0.69518  0.91940  0.00000
```

forms a lower triangular matrix from a random 4 by 4 matrix, omitting the main diagonal, and

```octave:13> tril (rand (4), -1)
ans =

0.06170  0.51396  0.00000  0.00000
0.96199  0.11986  0.35714  0.00000
0.16185  0.61442  0.79343  0.52029
0.68016  0.48835  0.63609  0.72113
```

forms a lower triangular matrix from a random 4 by 4 matrix, including the main diagonal and the first super-diagonal.