4.1. The commutator between the derivative and coordinate matrices

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] which will represent the coordinate partition

$$\mathbf{Q}\_{\mathcal{N}} \coloneqq \text{diag}\left(q\_1, q\_2, \dots, q\_N\right). \tag{55}$$

<sup>S</sup><sup>N</sup> <sup>¼</sup> <sup>1</sup> <sup>z</sup> � <sup>1</sup> z

Its determinant is

g ¼ g1; g2;…; gN � �<sup>T</sup>

edge points

⋮

ð Þ <sup>S</sup>N<sup>g</sup> <sup>1</sup> <sup>¼</sup> <sup>1</sup>

<sup>z</sup> � <sup>1</sup>=<sup>z</sup> z g<sup>1</sup> <sup>þ</sup><sup>X</sup>

<sup>z</sup> � <sup>1</sup>=<sup>z</sup> <sup>g</sup><sup>1</sup> <sup>þ</sup><sup>X</sup>

4

ð Þ <sup>S</sup>N<sup>g</sup> <sup>N</sup> <sup>¼</sup> <sup>1</sup>

4

ð Þ <sup>S</sup>N<sup>g</sup> <sup>2</sup><sup>j</sup> <sup>¼</sup> <sup>1</sup>

ð Þ <sup>S</sup>N<sup>g</sup> <sup>2</sup>jþ<sup>1</sup> <sup>¼</sup> <sup>1</sup>

exponential function evq.

BBBBBBBBBBBBBBBBBB@

0

1 �z 1 �z 1 … �z 1 z �1 1=z �1 1=z … �1 1=z 1 �1=z 1 �z 1 … �z 1 z �1 z �1 1=z … �1 1=z

1

http://dx.doi.org/10.5772/intechopen.74356

Matrices Which are Discrete Versions of Linear Operations

CCCCCCCCCCCCCCCCCCA

ð Þ vΔ : (60)

� � " #, (61)

<sup>g</sup>2kþ<sup>1</sup> <sup>z</sup> � <sup>g</sup>2<sup>k</sup>

<sup>g</sup>2kþ<sup>1</sup> � z g2<sup>k</sup>

� � " #, (64)

3

3

5, (62)

5, (63)

, (59)

33

1 �1=z 1 �1=z 1 … �z 1 z �1 z �1 z … �1 1=z 1 �1=z 1 �1=z 1 … �1=z 1 z �1 z �1 z … �1 1=z 1 �1=z 1 �1=z 1 … �1=z 1

<sup>∣</sup>SN<sup>∣</sup> <sup>¼</sup> sinh<sup>N</sup>�<sup>1</sup>

<sup>z</sup> � <sup>1</sup>=<sup>z</sup> <sup>g</sup><sup>1</sup> <sup>þ</sup><sup>X</sup>

j�1

k¼1

j

k¼1

<sup>z</sup> � <sup>1</sup>=<sup>z</sup> <sup>g</sup><sup>1</sup> <sup>þ</sup><sup>X</sup>

6. Transformation between coordinate and derivative representations

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

Since one of the eigenvalues of the derivative matrix is a continuous variable, we can talk of conjugate functions with a continuous argument v. The relationship between discrete vectors

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

M

i¼1

� � <sup>2</sup>

� � <sup>2</sup>

z g2kþ<sup>1</sup> � <sup>g</sup>2<sup>k</sup> � � þ<sup>X</sup>

<sup>g</sup>2kþ<sup>1</sup> � <sup>g</sup>2<sup>k</sup> z � � <sup>þ</sup> <sup>X</sup>

M

i¼1

, we obtain formulas for the finite differences integration, including the

<sup>g</sup>2iþ<sup>1</sup> � z g2<sup>i</sup>

M

k¼j

M

k¼jþ1

<sup>g</sup>2iþ<sup>1</sup> � <sup>g</sup>2<sup>i</sup> z

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

$$[\mathbf{D}\_N, \mathbf{Q}\_N] = \frac{\Delta}{2\chi(v, \Delta)} \begin{pmatrix} 0 & 1 & 0 & 0 & \dots & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & \dots & 0 & 0 & 0 \\ \vdots & & & & & & \\ 0 & 0 & 0 & 0 & \dots & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & \dots & 0 & 1 & 0 \\ \end{pmatrix} . \tag{56}$$

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

$$[\mathbf{D}\_{N\prime}\mathbf{Q}\_N] \approx I\_{\prime} \tag{57}$$

where I is the identity matrix, with the first and last elements replace with 1/2. Thus, coordinate and derivative matrices are finite differences conjugate of each other.
