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The Theodorus Spiral，Diagonal Matrices, Vector Algebra, and System Monitoring (Part 1)

T. Jay Bai, Ph.D., 615 Joanne St., Fort Collins, CO 80524, USA

tjbmdsm@yahoo.com, 1-970-495-9716

**Abstract: **

The "Theodorus Spiral" can be expressed as a 17*17 Diagonal Matrix. A diagonal matrix is a Multiplicative Group. After all zeroes are removed, the 17*17 diagonal matrix becomes a 17-component vector. The 17-Vector is still a Multiplicative Group, provided it defines a component-wise Multiplication. Vector Algebra offers a solution to our System Trend Analysis, and can be applied to investment management in a 17-financial market, as well as to other System Monitoring programs.

According to Wikipedia ^{[1]}, about 2,500 years ago, in order to prove that the square root of non-square numbers are irrational numbers, Theodorus of Cyrene (5th-century BC) proposed the "Theodorus Spiral". A two-dimensional Theodorus Spiral is demonstrited in the following figures:

This is a spiral that increment in outer wheel is：=1，the increment in diameter is: ，

and the rotation angle is: , where *i* = 1, 2, ..., 17.

From the perspective of modern mathematics, the "Theodorus Spiral" can be expressed in a 17 dimensional space, 17-hyperspace. As shown in the following figure, it is a fragment of the Theodorus Spiral in the 17-hyperspace:

Where, the origin is O, and the three end points in the fragment are represented by A, B, and C, respectively.

Two right-angled triangles ⊿OAB and ⊿OBC have a common side, OB.

and，, |AB|=1, |BC|=1,

| OA|=，|OB|=，|OC|=

As known，⊿OBC is a right-angled triangle,

However，according to the parallelogram rule,

so, we can launch

thus, ⊥⊿OAB plane.

Or, in the 17-hyperspace, perpendicular to the vertex B of the right-angled triangle ⊿OAB, intercept BC = 1, and then connect CO to get |OC|= .

Either way, it stands that the two planes, ⊿OAB and ⊿OBC, are perpendicular to each other.

In a two-dimensional plane, from the first figure, the edge 1 is perpendicular to the hypotenuse; but in the 17-hyperspace, the second figure above, the edge 1 is perpendicular to the planes defined by ，and ，while *i* = 1, 2, ..17.

Thus，all of the seventeen ’s=1 are perpendicular to each other in the 17-hyperspace. From the second figure, while the edge increases 1, the chord (diameter) length increases ( - ), at the same time, it rotates .

Therefore, it is a spiral, and it is the "Theodorus Spiral" in a 17-hyper space. In Theodorus' time, the Spiral was limited to 2π in a two-dimensional plane. The “Theodorus Spiral” in history rotated 17 times but stopped extension on . Our discussion today will be limited to 17-hyperspace as well, although we know that this Spiral can rotate in hyperspace infinitely.

Since the seventeen ’s=1 are mutually perpendicular to each other in the 17-hyperspace, thus they form a 17*17 Diagonal Matrix. We named it "Theodorus-Matrix", or "T-Matrix". The diagonal elements of this 17*17 "T-Matrix" equal 1, and the non-diagonal elements equal zeros:

T-Matrix=

1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0, 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0, 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0, 1,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0, 1,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0, 1,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0, 1,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0,0, 1,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1, 0

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 1

This diagram, a diagonal matrix consisting of 1 and 0, is called the "Identity Matrix", which is a matrix of multiplicative identity, and has a few special properties:

1) It can be multiplied by the left or right by a same dimensional matrix, but remains the same:

I*A = A*I = A

2) If the product of two matrices A and B is an Identity Matrix, these two matrices are mutually inverse to each other:

If A*B = I, then

A = B^{-1}, and B = A^{-1}

If the diagonal elements of the "T-matrix" are replaced by non zero real numbers, then it becomes a normal diagonal matrix and a normal diagonal matrix is a Multiplicative Group. Then the "T-matrix" has some practical applications. For example, if we replaced the diagonal elements with 17 daily closing prices of some funds, such as: DIA, FBALX,… FMCSX, then we formed a Market Diagonal Matrix that expresses the 17-fund market on a given day:

The Market Diagonal Matrix=

DIA, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0, FBALX,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0, FBIOX,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0, FBMPX,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0, FCNTX,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0, FCYIX,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0, FDCPX,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0, FDLSX,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0, FEQIX,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0, FFNOX,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0, FGRTX,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0, FIEUX,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0, FFNOX,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0, FIUIX,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0,0, FCLSX,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, FLVEX, 0

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, FMCSX

The diagonal elements of the matrix are the closing prices of 17 selected funds on a given day. For example, the first diagonal element is the daily closing price of fund that coded as DIA, the second is FBALX, and so on.

Furthermore, this Market Diagonal Matrix needs to be Standardized, i.e., divide each singal fund by the market. The market is expressed by the Square root of the Sum of the Square (SSS, or Vector sum) of the 17 funds. The SSS is named ShangGao Index, SGI^{[2]}. This procedure of dividing funds by market is called "Standardization", and the standardized matrix we name it System State Matrix. It (System State Matrix) looks like this:

DIA/SGI, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0, FBALX/SGI,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0, FBIOX/SGI,0,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0, FBMPX/SGI,0,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0, FCNTX/SGI,0,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0, FCYIX/SGI,0,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0, FDCPX/SGI,0,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0, FDLSX/SGI,0,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0, FEQIX/SGI,0,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0, FFNOX/SGI,0,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0, FGRTX/SGI,0,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0, FIEUX/SGI,0,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0, FFNOX/SGI,0,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0, FIUIX/SGI,0,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0,0, FCLSX/SGI,0,0

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, FLVEX/SGI, 0

0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, FMCSX/SGI

And the SGI=, the square root of the sum of the 17 diagonal elements squared.

Since the diagonal matrix is a Multiplicative Group, the System State Matrix (S) does have an inverse.

S^{-1}=1/s_{(i,i)},

the inverse of System State Matrix is a diagnol matrix, where *i*=1, 2,…17, provided that none of the closing price is zero, i.e. no funds are free.

After indexed SSM with time k, we can define the quotients of State of previous (k-1) and present (k) as the System Trend Vector to express the changing trends of the market, and each/every fund ^{[2]}:

Trend_{(i,i,k)}=S_{(i,i,k)}/S_{(i,i, k-1)}

where the Trend, and S are 17-diagonal matrices, the S' are system state matrices, the first and second subscripts indicating the row and the column, but the third subscript indicating time. Since they are 17-diagonal matrix, the second subscript can be omitted:

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)}

where, the first subscript "i " is indicating the funds, but the second subscript "k" is indicating time, and the funds are standardized:

S=S_{(i,k)}=fund'_{(i, k)}=fund_{(i, k)}/SGI_{(k)}, where *i*=1,2,...17

The expectation of the trend values are 1:

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)} =1, where *i*=1,2,...17, it means that there was no change. If

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)} >1, where *i*=1,2,...17, it means that the given fund is increasing, but

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)} <1, where *i*=1,2,...17, it means that the given fund is decreasing.

Furthermore, after we pick up three consecutive times of (k-1), k, and (k+1) from the market time series, we define trends:

T_{(k)} =k/(k-1), and T_{(k+1)}=(k+1)/k, as well as the second degree trends (TT) using the quotient of the trends:

TT=T_{(k+1)}/ T_{(k)}

=[(k+1)/k]/[k/(k-1)]

= [(k+1)*(k-1)]/[k^{2}],

which means that the second degree trend compares the two ending points with the middle point to express the turning points of the market movement in the 17-hyperspace.

The expectation values of the TT are 1:

TT_{(k+1)} = [(k+1)*(k-1)]/[k^{2}] =1, it showes that the product of the two ending points is equal to the quared middle point, there is no turning point, straight; while

TT_{(k+1)} = [(k+1)*(k-1)]/[k^{2}] > 1, it means that the product of the two ending points is higher than the quared middle point, so the turning point exists, and is concave;

but TT_{(k+1)} = [(k+1)*(k-1)]/[k^{2}] < 1, it means that the product of the two ending points is less than the squared middle point, so the turning point exists, and is convex.

Combining the three trends: up, even, or down, with second degree trends: concave, straight, or convex, these nine states describes each of the 17 funds and market more clearly than using real numbers could do before. Then the 17 funds can be put in order by their Trend Values and/or Second Degree Trend Values, to help investors make decisions, as we suppose that the market systems have some inertia in their neighborhood.

17-Component Vectors

The 17*(17-1) off diagnal zeros of the Theodorus Matrix are dummy elements that are not involved in above discussions. To save space, and to make functions/relations simpler and clearer,we deleted all the (17*16) zeros located on off diagonal position, then the T-Matrix becomes a 17-component Theorodus Vector:

Theodorus Vector =（1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1).

For the Vectors, we took the definition as: "Vectors are the variables with Magnitudes and Directions". In the following discussions, this T-Vector is handled as a point in 17-hyperspace, instead of 17*1 matrix. The magnitude of the T-Vector,or length, is SSS of the entire 17 components: |T-Vector|= ，

the direction of T-Vector in 17-hyperspace is expressed by 17 ,

that are 17 arctan values:（0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25）.

We repeate our discussion on Vector Algebra just as we discussed on Diagonal Matrices above: After replcing the 1 with the closing prices of the 17 funds, the T-Vector become 17-Market Vector, where the components of the vector are the closing prices of 17 selected funds on a given day.

The 17-Market Vector

= (DIA, FBALX, FBIOX, FBMPX, FCNTX, FCYIX, FDCPX, FDLSX, FEQIX, FFNOX, FGRTX, FIEUX, FFNOX, FIUIX, FCLSX, FLVEX, FMCSX)

For example, the first component is DIA, and the second is FBALX, and so on. The 17-market is defined as a 17-hyperspace where each/every fund functions as a dimension of the hyperspace. And a point/identity in the 17-hyperspace is expressed as a 17-vector. A 17-vector is defined by the coordinates of the vector' ending point in the 17-hyperspace,

The identities in the 17-hyperspace should be projected on to unit hypersphere., ie., divided by its vector length. This procedure is named Standardization. Standardization is to divide each singal fund by the market, which is expressed by Square root of the Sum of the Square (SSS, or Vector Length) of the 17 funds. The SSS was named as the ShangGao Index (SGI) mentioned above, and the standardized market vector is named System State Vector:

The 17-System State Vector =Market Vector' = Market Vector/SGI

_{}= (DIA/SGI, FBALX/SGI, FBIOX/SGI, FBMPX/SGI, FCNTX/SGI, FCYIX/SGI, FDCPX/SGI, FDLSX/SGI, FEQIX/SGI, FFNOX/SGI, FGRTX/SGI, FIEUX/SGI, FFNOX/SGI, FIUIX/SGI, FCLSX/SGI, FLVEX/SGI, FMCSX/SGI)

and SGI=, the square root of the sum of the 17 closing prices of the funds squared. After divided by SGI, this 17-vector is called the System State Vector for the 17-market on a given day.

Given a component-wise definition of the inverse: Inverse of a component is the responding component of the inverse.

S^{-1}=1/S_{(i)}

and definition of a component-wise multiplication:

C_{(i)}=A_{(i)}*B_{(i)}

and definition of a component-wise division:

A/B=A_{(i)}/B_{(i)} , where *i*=1,2,...17.

Then the 17-Market State Vector is a Multiplicative Group, provided that none of the closing prices of the 17 funds is zero, funds are not for free.

The vector can be expressed as bold upercase charactor:

**S=**S_{(i)} **, **and it can add a second indix for time, k. So (k-1), k, (k+1) indicates: previous, present, and next time, respectively.

We also can define the quotients of previous and present state, as System Trend Vector to express the systems' changing trends:

Trend_{(i,k)}= **S**_{(i,k)}/**S**_{(i, k-1)}

where, the first subscript "i " is indicating the funds, but the second is indicating time, and the market is standardized:

S_{(i, k)}=Market'=Market_{(i, k)}/SGI_{(k)}, where *i*=1,2,...17

The expectation of the trend values are 1:

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)} =1, where *i*=1,2,...17, it means that there is no change

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)} >1, where *i*=1,2,...17, it means that the given fund is increasing, but

Trend_{(i,k)}=S_{(i,k)}/S_{(i, k-1)} <1, where *i*=1,2,...17, it means that the given fund is decreasing.

Furthermore, after we pick up three consequtive times from the market, then it forms the time series of(k-1), k, and (k+1), we can define the second degree trends, it is trends of trends:

Since T_{(k)} =k/(k-1), and T_{(k+1)}=(k+1)/k, we can define the Second Degree Trends as the quotient of the trends:

TT=T_{(k+1)}/ T_{(k)}

=[(k+1)/k]/[k/(k-1)]

= [(k+1)*(k-1)]/[k^{2}],

The second degree trend compares the two ending points with the middle point, to show the turning points in the time series.

The expectation values of the TT are 1:

TT_{(k+1)} = [(k+1)*(k-1)]/[k^{2}] =1, it showes that there is no turning, straight;

while TT_{(k+1)} = [(k+1)*(k-1)]/[k^{2}] > 1, it means that the product of the two ending points is higher than the squared middle point, so the turning exists, and is concave; but, if

TT_{(k+1)} = [(k+1)*(k-1)]/[k^{2}] < 1, it means that the product of the two ending points is less than the squared middle point, so the turning exists, but is convex.

Combining the three trends: up, even, or down, with the second degree trends: concave, straight, or convex, these nine states describes each and every one of the 17 funds, as well as the market, simutaneously, and more clearly than using real numbers could do before. Then these 17 funds can be put in order by their Trend Values and/or second degree trend values, to help the investors make decisions, as we supposed that the market systems have some inertia in their neighbor-hood.

Summary of Part One**：**

Theodorus Spiral

The famous and ancient Theodorus Spiral has been expressed as a Spiral in a two-dimensional plane:

The increment in outer wheel is：=1，

the increment in diameter is: ，

and the rotation angle is: , or , where *i* = 1, 2, ..., 17.

**Diagonal Matrices**

The Spiral can also be expressed as an 17*17 Identity Matrix. Using 17 non zero real numbers replaces the diagonal elements, the Theodorus Spiral becomes a 17*17 diagonal matrix. A diagonal Matrix has inverse, and it is a Multiplicative Group.

**17-Vectors**

The map of the “Theodorus Spiral”in 17-hyperspace is a 17-component vector, and all its 17 components are 1:

Theodorus Vector =（1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1，1), and

the vector length is: Vector length= ，

the 17-direction of the T-Vector in hyperspace is expressed by 17 ,

that is 17-arctan=（0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25，0.25). The 17-Vector is an Identity Vector in 17-hyperspace. Vector is Multiplication Group, provided a component-wise definition of multiplication. After the components replaced by 17 funds, it becomes a market vector, can be used for trend analysis.

**Lookforward to Part Two:**

An application of vector algebra on investment market (to be continued) by MDSM Colorado Group

[1] Wikipedia，Spiral of Theodorus

[2] 超球面模型应用于股票排序的探讨

[3] Multi-Dimensional Sphere Model and Instantaneous Vegetation Trend Analysis

[4] 多维球面模型应用于草原监测数据分析的探讨

[6] 逆运算推动數集扩展

[5] 多元向量乘法群

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