TOEPLITZ Function :: SAS/IML(R) 12.1 User's Guide

TOEPLITZ Function

TOEPLITZ (a) ;

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 $(np) \times p$ or $p \times (np)$ matrix; the value returned is the $(np) \times (np)$ result.

The TOEPLITZ function uses the first $p \times p$ submatrix, $\mb {A}_1$ , of the argument matrix as the blocks of the main diagonal. The second $p \times p$ submatrix, $\mb {A}_2$ , of the argument matrix forms one secondary diagonal, with the transpose $\mb {A}_2^{\prime }$ forming the other. The remaining diagonals are formed accordingly. If the first $p \times p$ submatrix of the argument matrix is symmetric, the result is also symmetric. If $\mb {A}$ is $(np) \times p$ , the first columns of the returned matrix, $\mb {R}$ , are the same as $\mb {A}$ . If $\mb {A}$ is $p \times (np)$ , the first rows of $\mb {R}$ are the same as $\mb {A}$ .

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.

$\mb {A} = \left[ \mb {A}_1 | \mb {A}_2 | \mb {A}_3 |\cdots | \mb {A}_ n \right]$

and if $\mb {R}$ is the matrix formed by the TOEPLITZ function, then

$\mb {R} = \left[ \begin{array}{ccccccccc} \mb {A}_1 & | & \mb {A}_2 & | & \mb {A}_3 & | & \cdots & | & \mb {A}_ n \\ \mb {A}_2^{\prime } & | & \mb {A}_1 & | & \mb {A}_2 & | & \cdots & | & \mb {A}_{n-1} \\ \mb {A}_3^{\prime } & | & \mb {A}_2^{\prime } & | & \mb {A}_1 & | & \cdots & | & \mb {A}_{n-2} \\ \vdots & & & & & & \\ \mb {A}_ n^{\prime } & | & \mb {A}_{n-1}^{\prime } & | & \mb {A}_{n-2}^{\prime } & | & \cdots & | & \mb {A}_1 \end{array} \right]$

$\mb {A} = \left[ \begin{array}{c} \mb {A}_1 \\ \mb {A}_2 \\ \vdots \\ \mb {A}_ n \end{array} \right]$

and if $\mb {R}$ is the matrix formed by the TOEPLITZ function, then

$\mb {R} = \left[ \begin{array}{ccccccccc} \mb {A}_1 & | & \mb {A}_2^{\prime } & | & \mb {A}_3^{\prime } & | & \cdots & | & \mb {A}_ n^{\prime } \\ \mb {A}_2 & | & \mb {A}_1 & | & \mb {A}_2^{\prime } & | & \cdots & | & \mb {A}_{n-1}^{\prime } \\ \vdots & & & & & & \\ \mb {A}_ n & | & \mb {A}_{n-1} & | & \mb {A}_{n-2} & | & \cdots & | & \mb {A}_1 \end{array} \right] ~$

Three examples follow:

r1 = toeplitz(1:5);
r2 = toeplitz({1 2 ,
               3 4 ,
               5 6 ,
               7 8});
r3 = toeplitz({1 2 3 4,
               5 6 7 8});
print r1, r2, r3;

Figure 23.353: Toeplitz Matrices

r1
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

r2
1	2	5	7
3	4	6	8
5	6	1	2
7	8	3	4

r3
1	2	3	4
5	6	7	8
3	7	1	2
4	8	5	6