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.
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.
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.