### 21.1 Creating and Manipulating Diagonal/Permutation Matrices

A diagonal matrix is defined as a matrix that has zero entries outside the main diagonal; that is, `D(i,j) == 0` if `i != j`. Most often, square diagonal matrices are considered; however, the definition can equally be applied to non-square matrices, in which case we usually speak of a rectangular diagonal matrix.

A permutation matrix is defined as a square matrix that has a single element equal to unity in each row and each column; all other elements are zero. That is, there exists a permutation (vector) `p` such that `P(i,j) == 1` if `j == p(i)` and `P(i,j) == 0` otherwise.

Octave provides special treatment of real and complex rectangular diagonal matrices, as well as permutation matrices. They are stored as special objects, using efficient storage and algorithms, facilitating writing both readable and efficient matrix algebra expressions in the Octave language. The special treatment may be disabled by using the functions optimize_diagonal_matrix and optimize_permutation_matrix.

: val = optimize_diagonal_matrix ()
: old_val = optimize_diagonal_matrix (new_val)
: optimize_diagonal_matrix (new_val, "local")

Query or set whether a special space-efficient format is used for storing diagonal matrices.

The default value is true. If this option is set to false, Octave will store diagonal matrices as full matrices.

When called from inside a function with the `"local"` option, the setting is changed locally for the function and any subroutines it calls. The original setting is restored when exiting the function.

: val = optimize_permutation_matrix ()
: old_val = optimize_permutation_matrix (new_val)
: optimize_permutation_matrix (new_val, "local")

Query or set whether a special space-efficient format is used for storing permutation matrices.

The default value is true. If this option is set to false, Octave will store permutation matrices as full matrices.

When called from inside a function with the `"local"` option, the setting is changed locally for the function and any subroutines it calls. The original setting is restored when exiting the function.

```x = diag (rand (10, 1));