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 Language Reference

TOEPLITZ Function

generates a Toeplitz or block-Toeplitz matrix

TOEPLITZ( a)

where a is either a vector or a numeric matrix.

The TOEPLITZ function generates a Toeplitz matrix from a vector, or a block Toeplitz matrix from a matrix. A block Toeplitz matrix has the property that all matrices on the diagonals are the same. The argument a is an (np) ×p or p ×(np) matrix; the value returned is the (np) ×(np) result.

The TOEPLITZ function uses the first p ×p submatrix, A1, of the argument matrix as the blocks of the main diagonal. The second p ×p submatrix, A2, of the argument matrix forms one secondary diagonal, with the transpose A2' forming the other. The remaining diagonals are formed accordingly. If the first p ×p submatrix of the argument matrix is symmetric, the result is also symmetric. If A is (np) ×p, the first p columns of the returned matrix, R, will be the same as A. If A is p ×(np), the first p rows of R will be the same as A. The TOEPLITZ function is especially useful in time-series applications, where the covariance matrix of a set of variables with its lagged set of variables is often assumed to be a block Toeplitz matrix.

If
and if R is the matrix formed by the TOEPLITZ function, then
If
and if R is the matrix formed by the TOEPLITZ function, then
Three examples follow.
```    r=toeplitz(1:5);
```
```             R             5 rows      5 cols    (numeric)

1         2         3         4         5
2         1         2         3         4
3         2         1         2         3
4         3         2         1         2
5         4         3         2         1

r=toeplitz({1 2 ,
3 4 ,
5 6 ,
7 8});

R             4 rows      4 cols    (numeric)

1         2         5         7
3         4         6         8
5         6         1         2
7         8         3         4

r=toeplitz({1 2 3 4 ,
5 6 7 8});

R             4 rows      4 cols    (numeric)

1         2         3         4
5         6         7         8
3         7         1         2
4         8         5         6
```

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