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# 18. 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.

18.1 Finding Elements and Checking Conditions |  |
18.2 Rearranging Matrices |  |
18.3 Special Utility Matrices |  |
18.4 Famous Matrices |  |

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## 18.1 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, dim)
  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 ]

  If the optional argument dim is supplied, work along dimension dim. For example,

      any (eye (2, 4), 2)
           => [ 1; 1 ]

Built-in Function: all (x, dim)
  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 along dimension dim of a matrix, are nonzero.

Since the comparison operators (see section 10.4 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 its 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))`.

Mapping Function: xor (x, y)
  Return the `exclusive or' of the entries of x and y. For boolean expressions x and y, `xor (x, y)` is true if and only if x or y is true, but not if both x and y are true.

Function File: is_duplicate_entry (x)
  Return non-zero if any entries in x are duplicates of one another.

Function File: diff (x, k, dim)
  If x is a vector of length n, `diff (x)` is the vector of first differences x(2) - x(1), ..., x(n) - x(n-1).

  If x is a matrix, `diff (x)` is the matrix of column differences along the first non-singleton dimension.

  The second argument is optional. If supplied, `diff (x, k)`, where k is a nonnegative integer, returns the k-th differences. It is possible that k is larger than then first non-singleton dimension of the matrix. In this case, `diff` continues to take the differences along the next non-singleton dimension.

  The dimension along which to take the difference can be explicitly stated with the optional variable dim. In this case the k-th order differences are calculated along this dimension. In the case where k exceeds `size (x, dim)` then an empty matrix is returned.

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

      isinf ([13, Inf, NA, NaN])
           => [ 0, 1, 0, 0 ]

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

      isnan ([13, Inf, NA, NaN])
           => [ 0, 0, 0, 1 ]

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

      finite ([13, Inf, NA, NaN])
           => [ 1, 0, 0, 0 ]

Loadable Function: find (x)
  Return 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 nonzero values. For example,

      [i, j, v] = find (3 * eye (2))
           => i = [ 1; 2 ]
           => j = [ 1; 2 ]
           => v = [ 3; 3 ]

Function File: [err, y1, ...] = common_size (x1, ...)
  Determine if all input arguments are either scalar or of common size. If so, err is zero, and yi is a matrix of the common size with all entries equal to xi if this is a scalar or xi otherwise. If the inputs cannot be brought to a common size, errorcode is 1, and yi is xi. 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.

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## 18.2 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

Due to the difficult of define which axis about which to flip the matrix `fliplr` only work with 2-D arrays. To flip N-D arrays use `flipdim` instead.

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.

Function File: flipdim (x, dim)
  Return a copy of x flipped about the dimension dim. For example

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

Function File: rot90 (x, n)
  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). 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)

  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.

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:

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

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

Built-in Function: horzcat (array1, array2, ..., arrayN)
  Return the horizontal concatenation of N-d array objects, array1, array2, ..., arrayN along dimension 2.

Built-in Function: vertcat (array1, array2, ..., arrayN)
  Return the vertical concatenation of N-d array objects, array1, array2, ..., arrayN along dimension 1.

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).

Built-in Function: ipermute (a, iperm)
  The inverse of the `permute` function. The expression

      ipermute (permute (a, perm), perm)

  returns the original array a.

Function File: reshape (a, m, n, ...)
Function File: reshape (a, siz)
  Return a matrix with the given dimensions whose elements are taken from the matrix a. The elements of the matrix are access 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.

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

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 negative, the dimensions of x are shifted to the left, with the leading dimensions circulated to the end. If n is positive, then the dimensions of x are shifted to the right, with the n 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))
      => [2, 3, 1]
      size (shiftdim (x, 1))
      => [1, 1, 2, 3]
      [b, ns] = shiftdim (x);
      => b =  [1, 1, 1; 1, 1, 1]
      => ns = 1

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 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, it which case the dictionary order of the strings is used.

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

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 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 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 formed 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)
  Return the vector obtained by stacking the columns of the matrix x one above the other.

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

  Prepends (appends) 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 (end) 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 dim argument is given, then operate along this dimension.

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## 18.3 Special Utility Matrices

Built-in Function: eye (x)
Built-in Function: eye (n, m)
Built-in Function: eye (..., class)
  Return an identity matrix. If invoked with a single scalar argument, `eye` returns a square matrix with the dimension specified. If you supply two scalar arguments, `eye` takes them to be the number of rows and columns. If given a vector with two elements, `eye` uses the values of the elements as the number of rows and columns, respectively. For example,

      eye (3)
           =>  1  0  0
               0  1  0
               0  0  1

  The following expressions all produce the same result:

      eye (2)
      ==
      eye (2, 2)
      ==
      eye (size ([1, 2; 3, 4])

  The optional argument class, allows `eye` to return an array of the specified type, like

      val = zeros (n,m, "uint8")

  For compatibility with MATLAB, calling `eye` with no arguments is equivalent to calling it with an argument of 1.

Built-in Function: ones (x)
Built-in Function: ones (n, m)
Built-in Function: ones (n, m, k, ...)
Built-in Function: ones (..., class)
  Return a matrix or N-dimensional array whose elements are all 1. The arguments are handled the same as the arguments for `eye`.

  If you need to create a matrix whose values are all the same, you should use an expression like

      val_matrix = val * ones (n, m)

  The optional argument class, allows `ones` to return an array of the specified type, like

      val = ones (n,m, "uint8")

Built-in Function: zeros (x)
Built-in Function: zeros (n, m)
Built-in Function: zeros (n, m, k, ...)
Built-in Function: zeros (..., class)
  Return a matrix or N-dimensional array whose elements are all 0. The arguments are handled the same as the arguments for `eye`.

  The optional argument class, allows `zeros` to return an array of the specified type, like

      val = zeros (n,m, "uint8")

Function File: repmat (A, m, n)
Function File: repmat (A, [m n])
  Form a block matrix of size m by n, with a copy of matrix A as each element. If n is not specified, form an m by m block matrix.

Loadable Function: rand (x)
Loadable Function: rand (n, m)
Loadable Function: rand (`"seed"`, x)
  Return a matrix with random elements uniformly distributed on the interval (0, 1). The arguments are handled the same as the arguments for `eye`. In addition, you can set the seed for the random number generator using the form

      rand ("seed", x)

  where x is a scalar value. If called as

      rand ("seed")

  `rand` returns the current value of the seed.

Loadable Function: randn (x)
Loadable Function: randn (n, m)
Loadable Function: randn (`"seed"`, x)
  Return a matrix with normally distributed random elements. The arguments are handled the same as the arguments for `eye`. In addition, you can set the seed for the random number generator using the form

      randn ("seed", x)

  where x is a scalar value. If called as

      randn ("seed")

  `randn` returns the current value of the seed.

The `rand` and `randn` functions use separate generators. This ensures that

    rand ("seed", 13);
    randn ("seed", 13);
    u = rand (100, 1);
    n = randn (100, 1);

and

    rand ("seed", 13);
    randn ("seed", 13);
    u = zeros (100, 1);
    n = zeros (100, 1);
    for i = 1:100
      u(i) = rand ();
      n(i) = randn ();
    end

produce equivalent results.

Normally, `rand` and `randn` obtain their initial seeds from the system clock, so that the sequence of random numbers is not the same each time you run Octave. If you really do need for to reproduce a sequence of numbers exactly, you can set the seed to a specific value.

If it is invoked without arguments, `rand` and `randn` return a single element of a random sequence.

The `rand` and `randn` functions use Fortran code from RANLIB, a library of fortran routines for random number generation, compiled by Barry W. Brown and James Lovato of the Department of Biomathematics at The University of Texas, M.D. Anderson Cancer Center, Houston, TX 77030.

Function File: randperm (n)
  Return a row vector containing a random permutation of the integers from 1 to n.

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

The functions `linspace` and `logspace` make it very easy to create vectors with evenly or logarithmically spaced elements. See section 4.2 Ranges.

Built-in Function: linspace (base, limit, n)
  Return a row vector with n linearly spaced elements between base and limit. The number of elements, n, must be greater than 1. The base and limit are always included in the range. If base is greater than limit, the elements are stored in decreasing order. If the number of points is not specified, a value of 100 is used.

  The `linspace` function always returns a row vector.

Function File: logspace (base, limit, n)
  Similar to `linspace` except that the values are logarithmically spaced from 10^base to 10^limit.

  If limit is equal to pi, the points are between 10^base and pi, not 10^base and 10^pi, in order to be compatible with the corresponding MATLAB function.

Built-in Variable: warn_neg_dim_as_zero
  If the value of `warn_neg_dim_as_zero` is nonzero, print a warning for expressions like

      eye (-1)

  The default value is 0.

Built-in Variable: warn_imag_to_real
  If the value of `warn_imag_to_real` is nonzero, a warning is printed for implicit conversions of complex numbers to real numbers. The default value is 0.

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## 18.4 Famous Matrices

The following functions return famous matrix forms.

Function File: hankel (c, r)
  Return the Hankel matrix constructed given the first column c, and (optionally) the last row r. If the last element of c is not the same as the first element of r, the last element of c is used. If the second argument is omitted, it is assumed to be a vector of zeros with the same size as c.

  A Hankel matrix formed from an m-vector c, and an n-vector r, has the elements

      H(i,j) = c(i+j-1),  i+j-1 <= m;
      H(i,j) = r(i+j-m),  otherwise

Function File: hilb (n)
  Return the Hilbert matrix of order n. The i, j element of a Hilbert matrix is defined as

      H (i, j) = 1 / (i + j - 1)

Function File: invhilb (n)
  Return the inverse of a Hilbert matrix of order n. This can be computed computed exactly using

                  (i+j)         /n+i-1\  /n+j-1\   /i+j-2\ 2
       A(i,j) = -1      (i+j-1)(       )(       ) (       )
                                \ n-j /  \ n-i /   \ i-2 /

              = p(i) p(j) / (i+j-1)

  where

                   k  /k+n-1\   /n\
          p(k) = -1  (       ) (   )
                      \ k-1 /   \k/

  The validity of this formula can easily be checked by expanding the binomial coefficients in both formulas as factorials. It can be derived more directly via the theory of Cauchy matrices: see J. W. Demmel, Applied Numerical Linear Algebra, page 92.

  Compare this with the numerical calculation of `inverse (hilb (n))`, which suffers from the ill-conditioning of the Hilbert matrix, and the finite precision of your computer's floating point arithmetic.

Function File: sylvester_matrix (k)
  Return the Sylvester matrix of order n = 2^k.

Function File: toeplitz (c, r)
  Return the Toeplitz matrix constructed given the first column c, and (optionally) the first row r. If the first element of c is not the same as the first element of r, the first element of c is used. If the second argument is omitted, the first row is taken to be the same as the first column.

  A square Toeplitz matrix has the form:

      c(0)  r(1)   r(2)  ...  r(n)
      c(1)  c(0)   r(1)  ... r(n-1)
      c(2)  c(1)   c(0)  ... r(n-2)
       .     ,      ,   .      .
       .     ,      ,     .    .
       .     ,      ,       .  .
      c(n) c(n-1) c(n-2) ...  c(0)

Function File: vander (c)
  Return the Vandermonde matrix whose next to last column is c.

  A Vandermonde matrix has the form:

      c(1)^(n-1) ... c(1)^2  c(1)  1
      c(2)^(n-1) ... c(2)^2  c(2)  1
          .     .      .      .    .
          .       .    .      .    .
          .         .  .      .    .
      c(n)^(n-1) ... c(n)^2  c(n)  1

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