Language Reference

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 p columns of the returned matrix, $\mb{R}$, are the same as $\mb{A}$. If $\mb{A}$ is $p \times (np)$, the first p 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.

If

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

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

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

If

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

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 24.410: 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