The formatting for the expressions in this module are currently being revised.

Transformation from a Riemann sum equation to
the function for area spectrum (A) is the eigenvalue:

The area spectrum can be stretched through array duplication across multipule dimensions forming a matrix. The matrices complimenting the eigenvalue have an equal number of values. For this instance of
λ
λ
, a matrix formulation is necessary for the Riemann ∆x and (A) arrays by duplicating the vertical elements so that the horizontal size of each matrix equals the number of inputs of the original array:

Δx=EigenvectorsSubscriptxnn
Δx
Eigenvectors
Subscript
x
nn

(2)
=Subscriptx11Subscriptx12⋯Subscriptx1nSubscriptx21Subscriptx22⋯Subscriptx2n⋮⋮⋱⋮Subscriptxn1Subscriptxn2⋯Subscriptxnn
Subscript
x
11
Subscript
x
12
⋯
Subscript
x
1
n
Subscript
x
21
Subscript
x
22
⋯
Subscript
x
2
n
⋮
⋮
⋱
⋮
Subscript
x
n1
Subscript
x
n2
⋯
Subscript
x
nn

(3)
A=EigenvectorsSubscript∂0xnn
A
Eigenvectors
Subscript
x
nn

(4)
=Subscript∂0x11Subscript∂0x12⋯Subscript∂0x1nSubscript∂0x21Subscript∂0x22⋯Subscript∂0x2n⋮⋮⋱⋮Subscript∂0xn1Subscript∂0xn2⋯Subscript∂0xnn
Subscript
x
11
Subscript
x
12
⋯
Subscript
x
1
n
Subscript
x
21
Subscript
x
22
⋯
Subscript
x
2
n
⋮
⋮
⋱
⋮
Subscript
x
n1
Subscript
x
n2
⋯
Subscript
x
nn

(5)The eigenvalue representing a squared value of the n number of observations is congruent to the input in either matrix. Thereby, the eigenfunction for the vectors and data set independent variable is possible.

ErrorBoxTextDataAx=λΔx
ErrorBox
TextData
Ax=λΔx

(6)When
ErrorBoxTextDataλ
ErrorBox
TextData
λ
is applied to the Riemann ∆x, find the
matrix transformation by the function of x.

SetAx(n21-1)((b−1a)n-1)
Set
Ax
n
2
1
-1
b
-1
a
n
-1

(7)
SetAn(b−1a)x-1
Set
A
n
b
-1
a
x
-1

(8)The area spectrum is calculated from a data set
using the same variables as the Riemann sum equation and solved for A
with f(x):

∑
x
=1nn(b−1a)x-1
x
1
n
n
b
-1
a
x
-1

(9)The resulting equation (5) represents all possible areas of a data set calculated by the Riemann sum equation.