Singular & Characteristic Structures of Similar Multisectoral System

Singular & Characteristic Structures

of Similar Multisectoral Systems

Adamou, N.1 & M. Ciaschini2

12th International Conference of Input-Output TechniquesNew York University, May 18 - 22, 1998

Section 5.3 Application of Mathematical Techniques Tuesday, May 19, Room #408

Abstract: Singular as well as eigen decomposition are used examining similar linearmultisectoral systems. Linear transformations examining the relations between the singular valuesof the Leontief and its similar inverse, as well as singular and characteristic values specify theintrinsic structures based on eigenvector relations. Four specific structures are examined. Thestructure that associates the common eigenvalues of the Leontief and its similar inverses to theirdifferent eigenvectors. Second examined structure is the one that relates eigen to singular values.Finally, the two alternative ways transform the singular values of the Leontief inverse into thesingular values of its similar inverse are examined. A pragmatic example demonstrates thetheoretical exploration.

1 Introduction

Similar multisectoral linear systems are analyzed combining singular and eigen value decomposition.Thus, intrinsic structural characteristics of the entailed linear transformations are formed by the pertinenteigenvectors.

Similarity transformation between two matrices lays its foundation on the change of a basis of a matrix.

Direct purchases A ↑[ ] as well as direct sales r B [ ] are defined on the space of interindustry transactions in

their proportion to sectoral gross output.3 The diagonal matrix of gross output and its inverse serve astransition matrices of the similarity transformation.4 The same similarity transformation relates matrices

I − A ↑[ ] to I −

r B [ ] as well as their inverses Z and G . Similar matrices are equivalent to each other,

having common determinants and eigenvalues. The eigenvalues may be real or complex, based upon therelation between the trace and determinant. The corresponding eigenvectors are different for matrices Zand G . All matrices of the purchase requirement perspective5 have the same eigenvectors, while theireigenvalues are linearly related. The same holds true for their similar matrices of the sale allocationstandpoint.6

Singular value decomposition was proposed as an alternative to the traditional multisectoral multiplier,and their associated orthogonal matrices offer a way to analyze the intersectoral structure. Matrices Zand G , although they share common eigenvalues, have different real and distinct singular values.Eigenvalues were suggested as an impact measurement as well. Yet, although both singular values andeigenvalues were assumed to measure impact, a comparative examination of singular and eigen valuesprovided diverge results. This leads to an exploration of the relations between singular and eigen valuesand their structural implications in impact analysis. Singular and eigen values are independent of anypermutation of the productive interdependent sectors, while the associated vectors of the appropriatedecomposition reflect the applied sectoral permutation. This result leads one to examine the vectorspaces in their association to singular and eigenvalues of the Leontief and its similar inverses. Such anexamination of the spaces and structures formed by the linear transformations based on singular and eigen

1 Aristotelian University of Thessaloniki, School of Law & Economics, Greece.2 University of Macerata, Faculty of Law, Italy.3 A ↑[ ] = X[ ] diag x( )[ ]−1

and r B [ ] = diag x( )[ ]−1 X[ ].

4 A ↑[ ] = diag x( )[ ]

r B [ ] diag x( )[ ]−1

.

5 A ↑[ ] , I − A ↑[ ] and Z = I − A↑[ ] −1.

6 r B [ ] ,

I −

r B [ ] , and

G = I −

r B [ ] .

Singular & Characteristic Structures of Similar Multisectoral Systems2

value decomposition of similar matrices illuminates the intrinsic structure of a multisectoral linearsystem. This paper examines the linear transformations that relate singular and eigen values, as well asthe two different sets of singular values of similar matrices. Figure 1-A shows the objective of the paper.

Figure 1-A

Common Eigenvalues of Leontief & its Similar Inverse

Singular Values of Leontief Inverse

Singular Values of Similar LeontiefInverse

Λ

SGSZ

Relationships among Eigen & Singular Values

The involved eigenvectors determine the structure of the above mentioned linear transformations,presented in Figure 1-B.

Figure 1-B

Structures Relating Eigen & Singular Values

Λ

SZ SG

Eigevectors of the Leontief Inverse & its similar associated to the eigenvalues in correspondence with the diag. gross output

m

zev zeu zsev zseu

yv yu

kv ku

Eigenvectors of the Leontief Inverse &its similar related to Orthogonal Matrices of the SVD

y relationship between Singular Values

through z structures

k relationship between Singular Valuesthrough the Orthogonal Matrices of the SVDand diag. gross output

The structural information used are presented in Figure 2. The Leontief and its similar inverse have thesame eigenvalues and different eigenvectors, while there are decomposed in different singular valueterms.

Adamou, & Ciaschini, 12th International Conference of Input-Output Techniques 3

Figure 2

l1

ln

O

SL1

SLn

SGn

SGn

O

O

0

0 0

0

0

0ez1 ezn eg1 egn

vz1 vzn uz1 uzn vg1 vgn u g1 ugn

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

L

L

L

L

L

L

Eigenvectors

of Leontief Inverse

Eigenvectors of

similar Leontief Inverse

Eigenvalues

Singular Values

Left Orthogonal

RightOrthogonal

Left Orthogonal

RightOrthogonal

Singular Values

Leontief Inverse

Similar Leontief Inverse

2 Methodology

The Leontief inverse matrix Z and its similar G have the same eigenvalues Λ , while their respectiveassociated eigenvectors EZ and EG are different from each other providing the following decomposition:

(1) Z = EZΛEZ−1 G = EG Λ EG

−1

The real symmetric matrices normal7 ZTZ[ ] and GTG[ ] guarantee real and distinct eigenvalues, while

matrices Z and G may have complex conjugate eigenvalues. The square root of eigenvalues of the

symmetric matrices are the singular values SZ and SG . The matrix decomposition of ZTZ[ ] and GTG[ ]in terms of their own eigenvalues SZ

2 & SG2 and their corresponding matrices of orthonormal eigenvectors

U Z & UG , is known as spectral decomposition or principal axis theorem

(2) ZTZ[ ] = UZTSZ

2 U Z[ ] GTG[ ] =

UG

T SG2 UG[ ] .

Given matrices Z & G , eigenvectors, U L & UG , and the reciprocal of the square roots of eigenvalues,

SL−1 and SG

−1, one may define matrices V L and VG as V Z = ZU Z S Z−1 and VG = GUG SG

−1. Then, the

singular value decomposition of matrices Z & G are:

(3) Z = VZSZUZT G = VG SG UG

T .

Matrices V Z , VG , U Z and UG are orthogonal.8

7 Normal is a matrix that possesses a complete set of orthonormal eigenvectors, Strang (1988), p. 311.8 Orthogonal matrix means that the product of the matrix to its transpose is the identity matrix, and its transposed equals its

inverse. The first statement says, in other words, that any column multiplied by its appropriate row yields one, and thevalue of a column element is the reciprocal of the value of the appropriate row element. This indicates that columns of an


The unitary matrices of eigenvectors U ZT and VG

T were called reference structure for final use and value

added respectively and V ZT and UG

T the reference structures of production and allocation for grossoutput. It was argued that the columns of the reference structure need to be taken into considerationinstead of the unitary multipliers.9

In a similar manner, one may identify the decomposition of the eigenvalues Λ of similar matrices.

Simplifying by µ[ ] = EZ−1 diag x( )EG[ ] , we have

(4) Λ = µ[ ]Λ µ−1[ ] .

Structure µ[ ] relates the inverse of eigenvectors of the Leontief inverse to the eigenvectors of its similarinverse through the diagonal of gross output. This decomposition of eigenvalues provides the structure ofthe interrelation of eigenvectors of similar matrices though the transition matrix of gross output.

The structures relating eigenvectors to the right and left orthogonal matrices of the singular value

decomposition are ζev[ ] = EZ−1V Z[ ] , ζeu[ ] = U Z

T EZ[ ] , ζevs[ ] = EG

−1V G[ ] , and ζeus[ ] = UG

T EG[ ] . These

structures transform the common eigenvalues into different singular values as:

(5) Λ = ζev[ ] SZ ζeu[ ] = ζevs[ ] SG ζeu

s[ ] .

Figure 3

E

V

-1

Z

Z

zev

z

SZ

eu

U

EZ

Z

T

S SL =Z

L z z=ev eu

s s

E

EV

U

G

-1 T

G

G

G

G

SG

This leads towards the transformation of the singular values of the similar matrix as:

SZ = V ZT EZ EG

−1( )VG[ ] SG UGT EG EZ

−1( )U Z[ ] ⇔ SZ = VZT η1( ) VG[ ]SG UG

T η1−1( )U Z[ ]

(6) SZ = ψ v[ ] SG ψu[ ]

where η[ ] = EZEG−1[ ] , ψv[ ] = ζev

−1ζevs[ ] , and ψu[ ] = ζeu

s ζeu−1[ ] . Structure η[ ] gives the direct relation of

eigenvectors of similar matrices, while structure µ[ ] diagonalizes this relation by incorporating thediagonal gross output.

orthogonal matrix are orthonormal. Orthonormal eigenvectors are orthogonal scaled to a unit length. Strang (1988),pp.296-297.

9 Ciaschini, M. (1993) op. cit., p. 146.


Figure 4

SZ

SG

VZ

T

EZ

EG

−1

VG

h

UG

T

EG

EZ

−1

UZ

−1h

= y yV U

SG

Combining similarity to the singular value decomposition of Z and G we have

SZ = V ZT diag x( )[ ]VG[ ] SG UG

T diag x( )[ ]−1 U Z[ ] , or:

(7) SZ = κ v[ ] SG κ u[ ]

where κv[ ] = V ZT diag x( )[ ]VG[ ] and κu[ ] = UG

T diag x( )[ ]−1 UZ[ ] .

Figure 5

SZ

SG

VZ

T VG

UG

TU

Z

= k kV U

SG

diag (x) diag (x)-1


Structure κv[ ] relates the left eigenspace of the singular value decomposition of similar matrices through

the diagonal of gross output. In the same way, structure κu[ ] relates the right eigenspace of the singularvalue decomposition of similar matrices through the diagonal of gross output.

Structures κv[ ] & ψv[ ] and κu[ ] & ψu[ ] , respectively, are equivalent.

3 Concluding Implications


Singular Values of the Leontief Inverse

R2 = 0.9602

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1955 1960 1965 1970 1975 1980 1985

Singular Values of the Similar Leontief Inverse

R2 = 0.979

0

2

4

6

8

10

12

14

16

1955 1960 1965 1970 1975 1980 1985

Eigenvalues of the Leontief Inverse

R2 = 0.9797

0.0

0.5

1.0

1.5

2.0

2.5

1955 1960 1965 1970 1975 1980 1985


Singular Values of the Leontief Inverse1955 1960 1965 1970 1975 1980 1985

1 3.04 3.19 3.00 2.75 2.96 3.22 2.912 1.16 1.16 1.12 1.14 1.21 1.17 1.163 1.13 1.10 1.07 1.08 1.15 1.12 1.104 1.10 1.06 1.06 1.08 1.09 1.08 1.085 1.03 1.05 1.05 1.03 1.05 1.04 1.076 1.01 1.04 1.04 1.02 1.04 1.03 1.027 1.00 1.02 1.01 1.02 1.01 1.02 1.028 1.00 1.00 1.00 1.00 1.00 0.99 1.009 0.99 1.00 1.00 0.99 0.99 0.99 0.9910 0.96 0.99 0.99 0.98 0.97 0.98 0.9811 0.95 0.96 0.97 0.96 0.90 0.96 0.9812 0.90 0.92 0.95 0.95 0.86 0.90 0.9313 0.67 0.73 0.73 0.80 0.77 0.75 0.76

Singular Values of the Similar Leontief Inverse1955 1960 1965 1970 1975 1980 1985

1 3.94 5.34 5.76 6.73 12.20 13.70 11.902 1.32 1.31 1.33 1.38 1.45 1.48 1.463 1.14 1.17 1.15 1.16 1.14 1.17 1.184 1.10 1.08 1.06 1.09 1.10 1.13 1.085 1.05 1.03 1.04 1.06 1.06 1.08 1.056 1.02 1.02 1.03 1.02 1.03 1.02 1.037 1.01 1.01 1.01 1.01 1.01 1.00 1.018 1.00 0.99 0.98 1.00 1.00 0.98 1.009 0.95 0.98 0.98 0.98 0.97 0.98 0.9810 0.94 0.94 0.97 0.97 0.94 0.96 0.9711 0.93 0.93 0.95 0.94 0.93 0.91 0.8912 0.79 0.83 0.82 0.79 0.81 0.78 0.8113 0.54 0.45 0.37 0.32 0.17 0.16 0.18

Eigenvalues of the Leontief Inverse1955 1960 1965 1970 1975 1980 1985

1 2.16 2.42 2.27 2.24 2.36 2.48 2.292 1.10 1.15 1.10 1.14 1.14 1.15 1.163 1.07 0.06 1.07 0.04 1.05 1.09 1.09 1.04 0.08 1.094 1.07 -0.06 1.07 -0.04 1.05 0.03 1.08 1.05 0.10 1.04 -0.08 1.085 1.04 1.06 1.05 -0.03 1.03 0.03 1.05 -0.10 1.03 0.01 1.03 0.056 1.02 1.03 1.03 1.03 -0.03 1.01 1.03 -0.01 1.03 -0.057 1.00 1.00 0.00 1.00 0.01 1.00 1.00 1.00 0.01 1.018 1.00 1.00 0.00 1.00 -0.01 1.00 0.99 0.02 1.00 -0.01 1.009 1.00 0.02 1.00 0.03 1.00 0.99 0.99 -0.02 1.00 0.02 0.98 0.02

10 1.00 -0.02 1.00 -0.03 0.99 0.04 0.98 0.98 0.09 1.00 -0.02 0.98 -0.0211 0.96 0.99 0.99 -0.04 0.97 0.98 -0.09 0.99 0.98 0.0112 0.95 0.03 0.94 0.01 0.98 0.97 0.05 0.95 0.02 0.97 0.06 0.98 -0.0113 0.95 -0.03 0.94 -0.01 0.96 0.97 -0.05 0.95 -0.02 0.97 -0.06 0.96


BIBLIOGRAPHY

Adamou N., (1997) “Impact Analysis Reexamined: Structural Implications for the USA,” discussion paper.Adamou N. (1996a) “Determinants, Eigenvalues & Eigenvectors in Similar Interindustry Systems,” discussion

paper.Adamou N. (1996b) “Similar Eigenstructure of the Irish Production & Allocation,” discussion paper.Adamou N. (1995) “Similarity Symmetrical Equivalencies between ‘demand’ - ‘supply’ aspects in an interindustry

system: Transformations, weighted multiplier decomposition & distributions” 11th InternationalConference on Input-Output Techniques, Delhi, India, 27 Nov. - 1 Dec.

Bródy A. (1970) Proportions Prices and Planning, Budapest, Akadémiai Kiadó.Ciaschini M. (1993) Modelling and Structure of the Economy, Chapman & Hall.Ciaschini M. (1989a) “Output structure and multiplier analysis” 18th Annual Meeting of the Northeast Regional

Science Association, Northeastern University, Boston.Ciaschini M. (1989b) “Scale and Structure in Economic Modelling,” Economic Modelling, 6 (4).Dowling E. (1980) Mathematics for Economists, Schaum's Outline Series, McGraw-Hill.Henry E. W. (1988) “Eigen-Vector Analysis of the Leontief Inverse - An Empirical Approach with Numerical

Illustration by 14 Sector Data,” 9th International Conference on Input-Output Techniques, Hungary,August-September.

Morishima M. (1964) Equilibrium, Stability and Growth: A Multisectoral Analysis, Oxford University Press.Strang G. (1988) Linear Algebra and its Applications, third Edition, HBJ College Publishers.Wong Y. K. (1954) “Some Mathematical Concepts for Linear Economic Models,” in Morgenstern (ed.) Economic

Activity Analysis, J. Wiley & Sons.Woodbury M. A. (1954) “Characteristic roots of Input-Output Matrices,” in Morgenstern (ed.) Economic Activity

Analysis, J. Wiley & Sons.


Descriptive Visual Appendix

Italian Three Sector Graphical Presentation of Matrix Structures

Leontief Inverse

Z =1.212 0.073 0.019

0.293 1.622 0.267

0.112 0.208 1.221

Similar Leontief Inverse

G =1.212 0.961 0.178

0.022 1.622 0.191

0.012 0.290 1.221

The roots of the characteristic polynomial of the Leontief inversef t( ) =− t3 + 4.056 t 2 − 5.352t + 2.309

are its eigenvalues.


Eigenvalues of the Leontief Inverse

Λ =1.77384 0 0

0 1.17253 0

0 0 1.1106

Eigenvectors of the Leontief Inverse

EZ =0.133 0.871 0.255

0.917 −0.405 −0.558

0.374 −0.275 0.788

Eigenvectors of the Leontief Similar Inverse

EG =−0.854 −0.998 0.937

−0.450 0.035 −0.157

−0.256 0.033 0.309

Singular Values of the Leontief Inverse

SZ =1.802 0 0

0 1.158 0

0 0 1.106


Singular Values of the Leontief Similar Inverse

SG =2.106 0 0

0 1.157 0

0 0 0.947

Left eigenvector of the Leontief's inverse Singular Value Decomposition

V Z =−0.276 0.910 0.307

−0.878 −0.110 −0.464

−0.389 −0.398 0.830

Right eigenvector of the Leontief's inverse Singular Value Decomposition

U ZT =

−0.353 −0.846 −0.396

0.886 −0.168 −0.430

0.298 −0.504 0.810

Left eigenvector of the Similar Leontief's inverse Singular Value Decomposition

VG =−0.644 −0.364 0.671

−0.707 −0.048 −0.705

−0.290 0.929 0.226


Right eigenvector of the Similar Leontief's inverse Singular Value Decomposition

UGT =

−0.380 −0.879 −0.287

−0.373 −0.138 0.917

0.845 −0.456 0.275

Diagonal Matrix of Gross Output

diag x( ) =0.727 0 0

0 9.487 0

0 0 6.810

Eigenvectors of the Leontief inverse multiplied to the inverse Eigenvectors of the Leontief similarinverse

η = EZEG−1[ ]

η =−0.789 −0.490 2.974

0.273 −0.816 −3.054

0.242 −1.802 0.900

µ = EZ−1diag x( )EG[ ]

µ =−4.663 0 0

0 −0.833 0

0 0 2.667


ζev = EZ−1V Z[ ]

ζev =−1.048 0.216 0.123

−0.143 1.079 0.037

−0.045 −0.230 1.007

ζeu = U ZT EZ[ ]

ζeu =−0.972 0.144 0.069

−0.197 0.959 −0.019

−0.119 0.240 0.997

ζevs = EG

−1VG[ ]

ζevs =

1.441 −0.463 1.010

−0.314 2.926 −0.055

0.291 2.304 1.577

ζeus = UG

T EG[ ]

ζeus =

0.795 0.338 −0.307

0.146 0.399 −0.045

−0.588 −0.851 0.950


ψ V

ψ V =−1.462 1.369 −0.833

−0.490 2.788 −0.214

0.110 2.988 1.478

κ V =−0.728 −0.174 0.676

1.211 3.777 −0.177

−4.395 −6.749 6.613

ψU

ψU =−0.899 0.547 −0.235

−0.241 0.457 −0.019

0.744 −1.220 0.877

κU =0.280 −0.429 −0.143

0.140 −0.510 −0.036

−0.386 1.021 0.403


Analytical Visual Appendix

Eigenvalues Diagonal of Gross Output

Λ =1.77384 0 0

0 1.17253 0

0 0 1.1106

diag x( ) =0.727 0 0

0 9.487 0

0 0 6.810

P1

P2

Ind.

Serv.

Agr.

1

2

3

P1

P2

Ind.

Serv.

Agr.1 2

3

Singular Values of the Leontief Inverse Singular Values of the Similar Leontief Inverse

SZ =1.802 0 0

0 1.158 0

0 0 1.106

SG =2.106 0 0

0 1.157 0

0 0 0.947

P1

P2

Col. 1

Col. 2

Col. 3

P1

P2

Col. 1

Col. 2

Col. 3


Leontief Inverse Similar Leontief Inverse

Z =1.212 0.073 0.019

0.293 1.622 0.267

0.112 0.208 1.221

G =1.212 0.961 0.178

0.022 1.622 0.191

0.012 0.290 1.221

Agricul

Industr

Service

P1

P2

P31

2

3

Agricul

Industr

Service

P1

P2

P3

1

2

3

EZ =0.133 0.871 0.255

0.917 −0.405 −0.558

0.374 −0.275 0.788

EG =−0.854 −0.998 0.937

−0.450 0.035 −0.157

−0.256 0.033 0.309

P1

P2

Service

Industry

Agricul

1

23

P1

P2

Service

Industry

Agricul

12

3


Direct Interaction of Eigenvectors of the Leontief & its Similar Inverses

η = EZEG−1[ ] η =

−0.789 −0.490 2.974

0.273 −0.816 −3.054

0.242 −1.802 0.900

Agricul

Industr

Service

P1

P2

P3

3

2

1

Interaction of Eigenvectors of the Leontief & its Similar Inverses through Gross Output

µ = EZ−1diag x( )EG[ ] µ =

−4.663 0 0

0 −0.833 0

0 0 2.667

Agricul

Industr

Service

P1

P2P3

1

2

3


V Z =−0.276 0.910 0.307

−0.878 −0.110 −0.464

−0.389 −0.398 0.830

VG =−0.644 −0.364 0.671

−0.707 −0.048 −0.705

−0.290 0.929 0.226

Agricul

Industr

Service

P1

P2

P3

1

3

2 Agricul

Industr

Service

P1

P2

P3 3

1

2

U ZT =

−0.353 −0.846 −0.396

0.886 −0.168 −0.430

0.298 −0.504 0.810

UGT =

−0.380 −0.879 −0.287

−0.373 −0.138 0.917

0.845 −0.456 0.275

Agricul

Industr

Service

P1

P2

P32

3

1

Agricul

Industr

Service

P1

P2

P32

3

1


ζev = EZ−1V Z[ ] ζev

s = EG−1VG[ ]

ζev =−1.048 0.216 0.123

−0.143 1.079 0.037

−0.045 −0.230 1.007

ζev

s =1.441 −0.463 1.010

−0.314 2.926 −0.055

0.291 2.304 1.577

Agricul

Industr

ServiceP1

P2

P31

23

Agricul

Industr

Service

P1

P2

P3

12

3

ζeu = U ZT EZ[ ] ζev

s = EG−1VG[ ]

ζeu =−0.972 0.144 0.069

−0.197 0.959 −0.019

−0.119 0.240 0.997

ζev

s =1.441 −0.463 1.010

−0.314 2.926 −0.055

0.291 2.304 1.577

Agricul

Industr

Service

P1

P2P3 1

23

AgriculIndustr

Service

P1

P2P3

12

3


κ V =−0.728 −0.174 0.676

1.211 3.777 −0.177

−4.395 −6.749 6.613

κU =0.280 −0.429 −0.143

0.140 −0.510 −0.036

−0.386 1.021 0.403

P1

P2Service

Industry

Agricul

1

2

3

P1

P2Service

Industry

Agricul

12

3

ψ V =−1.462 1.369 −0.833

−0.490 2.788 −0.214

0.110 2.988 1.478

ψU =−0.899 0.547 −0.235

−0.241 0.457 −0.019

0.744 −1.220 0.877

P1

P2

Service

Industry

Agricul

1

23

P1

P2

Service

Industry

Agricul

1

23


Eigenvalues Diagonal of Gross Output

Principal ComponentsEigenValue:Percent:CumPercent:Eigenvectors:Column 1Column 2Column 3

1.5000 50.0000 50.0000

0.40825 0.40825-0.81650

1.5000 50.0000

100.0000

-0.70711 0.70711 0.00000

0.0000 0.0000

100.0000

0.57735 0.57735 0.57735

Principal ComponentsEigenValue:Percent:CumPercent:Eigenvectors:AgricultureIndustryServices

1.5000 50.0000 50.0000

-0.40825-0.40825 0.81650

1.5000 50.0000

100.0000

-0.70711 0.70711 0.00000

-0.0000 -0.0000

100.0000

0.57735 0.57735 0.57735

0.707 -1.225 0.000 -0.707 -1.225 0.0000.707 1.225 0.000 -0.707 1.225 0.000

-1.414 0.000 0.000 1.414 0.000 0.000

Singular Values of the Leontief Inverse Singular Values of the Similar Leontief Inverse


1.5000 50.0000 50.0000

-0.70711 0.70711 0.00000

1.5000 50.0000

100.0000

0.40825 0.40825-0.81650

-0.0000 -0.0000

100.0000

0.57735 0.57735 0.57735


1.5000 50.0000 50.0000

-0.70711 0.70711 0.00000

1.5000 50.0000

100.0000

0.40825 0.40825-0.81650

-0.0000 -0.0000

100.0000

0.57735 0.57735 0.57735

-1.225 0.707 0.000 -1.225 0.707 0.0001.225 0.707 0.000 1.225 0.707 0.0000.000 -1.414 0.000 0.000 -1.414 0.000

Leontief Inverse Similar Leontief Inverse


1.7944 59.8121 59.8121

-0.73499 0.20076 0.64768

1.2056 40.1879

100.0000

-0.15949 0.87719-0.45289

-0.0000 -0.0000

100.0000

0.65905 0.43617 0.61270

Principal ComponentsEigenValue:Percent:CumPercent:Eigenvectors:AgriculureIndustryServices

2.0102 67.0066 67.0066

0.36693 0.60656-0.70530

0.9898 32.9934

100.0000

0.85841-0.51294 0.00546

-0.0000 -0.0000

100.0000

0.35847 0.60744 0.70889

Eigenvectors of the Leontief Inverse Eigenvectors of the Similar Leontief Inverse


2.2104 73.6794 73.6794

-0.67259 0.53198 0.51441

0.7896 26.3206

100.0000

0.00985-0.68864 0.72504

0.0000 0.0000

100.0000

0.73995 0.49272 0.45793


2.7217 90.7237 90.7237

0.56768 0.60582-0.55742

0.2783 9.2763

100.0000

0.66455 0.06243 0.74463

-0.0000 -0.0000

100.0000

-0.48591 0.79315 0.36716

Singular & Characteristic Structures of Similar Multisectoral System

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