| TOEPLITZ Function |
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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 
or 
matrix; the value returned is the 
result.
The TOEPLITZ function uses the first 
submatrix, 
, of the argument matrix as the blocks of the main diagonal. The second submatrix, 
, of the argument matrix forms one secondary diagonal, with the transpose 
forming the other. The remaining diagonals are formed accordingly. If the first submatrix of the argument matrix is symmetric, the result is also symmetric. If 
is , the first 
columns of the returned matrix, 
, are the same as . If is , the first rows of are the same as .
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
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and if is the matrix formed by the TOEPLITZ function, then
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If
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and if is the matrix formed by the TOEPLITZ function, then
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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