5. An integration matrix

Since the determinant of the derivative matrix D<sup>N</sup> is not always zero, we expect that there exist an inverse of it. At a local level, the inverse of the finite differences derivation is the summation as was found in Eq. (46). In this section, we determine the inverse of the derivative matrix, and we find that it is a global finite difference integration operation.

Once we know the eigenvalues and eigenvectors of the derivative matrix DN, it turns out that we also know the eigenvectors and eigenvalues of the inverse matrix, when it exists. In fact, the equality DNe<sup>m</sup> ¼ λmem, with λ<sup>m</sup> 6¼ 0, imply that

$$\mathbf{D}\_N^{-1} \mathbf{e}\_m = \lambda\_m^{-1} \mathbf{e}\_m. \tag{58}$$

The inverse matrix <sup>S</sup><sup>N</sup> <sup>¼</sup> <sup>D</sup>�<sup>1</sup> <sup>N</sup> is

#### Matrices Which are Discrete Versions of Linear Operations http://dx.doi.org/10.5772/intechopen.74356 33

$$\mathbf{S}\_{N} = \frac{1}{z - \frac{1}{z}} \begin{pmatrix} 1 & -z & 1 & -z & 1 & \dots & -z & 1 \\ z & -1 & 1/z & -1 & 1/z & \dots & -1 & 1/z \\ 1 & -1/z & 1 & -z & 1 & \dots & -z & 1 \\ z & -1 & z & -1 & 1/z & \dots & -1 & 1/z \\ \vdots \\ 1 & -1/z & 1 & -1/z & 1 & \dots & -z & 1 \\ z & -1 & z & -1 & z & \dots & -1 & 1/z \\ 1 & -1/z & 1 & -1/z & 1 & \dots & -1/z & 1 \\ z & -1 & z & -1 & z & \dots & -1 & 1/z \\ 1 & -1/z & 1 & -1/z & 1 & \dots & -1/z & 1 \\ \end{pmatrix} \tag{59}$$

Its determinant is

4.1. The commutator between the derivative and coordinate matrices

[QN] which will represent the coordinate partition

32 Matrix Theory-Applications and Theorems

½ �¼ DN; Q<sup>N</sup>

just

5. An integration matrix

The inverse matrix <sup>S</sup><sup>N</sup> <sup>¼</sup> <sup>D</sup>�<sup>1</sup>

The commutator between the partition and the finite differences derivative can also be calculated from a global point of view using the corresponding matrices. Let the diagonal matrix

QN≔diag q1; q2;…; qN

⋮

This is a kind of nearest neighbors' average operator, inside the interval. The small Δ limit is

where I is the identity matrix, with the first and last elements replace with 1/2. Thus, coordi-

Since the determinant of the derivative matrix D<sup>N</sup> is not always zero, we expect that there exist an inverse of it. At a local level, the inverse of the finite differences derivation is the summation as was found in Eq. (46). In this section, we determine the inverse of the derivative matrix, and

Once we know the eigenvalues and eigenvectors of the derivative matrix DN, it turns out that we also know the eigenvectors and eigenvalues of the inverse matrix, when it exists. In fact, the

<sup>N</sup> <sup>e</sup><sup>m</sup> <sup>¼</sup> <sup>λ</sup>�<sup>1</sup>

D�<sup>1</sup>

BBBBBBBBBBBBBBBB@

0100 … 000 1010 … 000 0101 … 000

0000 … 010 0000 … 101 0000 … 010

Then, the commutator between the derivative matrix and the coordinate matrix is

0

Δ 2χð Þ v;Δ

nate and derivative matrices are finite differences conjugate of each other.

we find that it is a global finite difference integration operation.

<sup>N</sup> is

equality DNe<sup>m</sup> ¼ λmem, with λ<sup>m</sup> 6¼ 0, imply that

� �: (55)

1

CCCCCCCCCCCCCCCCA

½ � DN, Q<sup>N</sup> ≈ I, (57)

<sup>m</sup> em: (58)

: (56)

$$|\mathbf{S}\_N| = \sinh^{N-1}(v\Delta). \tag{60}$$

This matrix represents an integration on the partition, with an exact value when it is applied to the exponential function evq on the partition. When applied to an arbitrary vector g ¼ g1; g2;…; gN � �<sup>T</sup> , we obtain formulas for the finite differences integration, including the edge points

$$(\mathbf{S}\_{\rm N}\mathbf{g})\_1 = \frac{1}{z - 1/z} \left[ \mathbf{g}\_1 + \sum\_{i=1}^{M} (\mathbf{g}\_{2i+1} - z\mathbf{g}\_{2i}) \right] \tag{61}$$

$$(\mathbf{S}\_{\mathbf{N}}\mathbf{g})\_{2j} = \frac{1}{z - 1/z} \left[ z\,\mathbf{g}\_1 + \sum\_{k=1}^{j-1} \left( z\,\mathbf{g}\_{2k+1} - \mathbf{g}\_{2k} \right) + \sum\_{k=j}^{M} \left( \frac{\mathbf{g}\_{2k+1}}{z} - \mathbf{g}\_{2k} \right) \right],\tag{62}$$

$$(\mathbf{S}\_{\mathbf{N}}\mathbf{g})\_{2j+1} = \frac{1}{z - 1/z} \left[ g\_1 + \sum\_{k=1}^{j} \left( g\_{2k+1} - \frac{g\_{2k}}{z} \right) + \sum\_{k=j+1}^{M} \left( g\_{2k+1} - z g\_{2k} \right) \right],\tag{63}$$

$$(\mathbf{S}\_N \mathbf{g})\_N = \frac{1}{z - 1/z} \left[ \mathbf{g}\_1 + \sum\_{i=1}^M \left( \mathbf{g}\_{2i+1} - \frac{\mathbf{g}\_{2i}}{z} \right) \right] \tag{64}$$

where N ¼ 2M þ 1. These are new formulas for discrete integration for the exponential function on a partition of equally separated points with the characteristic that it is exact for the exponential function evq.
