Linear Algebra 2

Linear Algebra 2 Matrix Eigenvalue Problems

References 

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Digeteo (2009). Scilab [Software]. Available at http://www. scilab.org/ Erwin Kreyszig, “Advanced Engineering Mathmatics”, 8th Edition, Copyright © 2003 John Wiley & Sons, Peter V. O’Neil, “Advanced Engineering Mathematics”, Copyright © 2007, Nelson, ad division of Thomson Canada Ltd. Strang, Gilbert. (Spring 2005). MIT OpenCourseWare. Linear Algebra. Video Lectured retrieved from http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLec Williams, Gareth. “Linear Algebra with applications”. 3rd Edition. Copyright © 1996. Times mirror Higher education Group, Inc.

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Let A = [ajk ] be a given n x n matrix and consider the vector equation (1) A value of λ for which (1) has solution x≠ 0 is called eigenvalue or characteristic value of the matrix A. The corresponding solution x≠ 0 of (1) is called eigenvectors or characteristic vectors The set of eigenvalues is called spectrum of A The largest of absolute value of eigenvalues of A is called the spectral radius of A. The set of all eigenvectors corresponding to an eigenvalue of A, together with 0, forms a vector space, called the eigenspace of A.

Ax = λx

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Equation

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Matrix

− 5 x1 + 2 x2 = λx1 2 x1 − 2 x2 = λx2

( − 5 − λ ) x1 + 2 x2 = 0 2 x1 + ( − 2 − λ ) x2 = 0 D( λ ) = ( − 5 − λ )( − 2 − λ ) − 4 = λ2 + 7λ + 6 = 0, λ1 = −1, λ2 = −6 4 x1 + 2 x2 = 0, x1 + 2 x2 = 0 2 x1 − x2 = 0,

2 x1 + 4 x2 = 0

x2 = 2 x1 ,

x 2 = − x1 / 2

− 5 2  A=  2 − 2    x1  − 5 2   x1  Ax =  = λ      2 − 2   x2   x2  2  − 5 − λ D(λ ) = det( A − λI ) =  − 2 − λ   2 x1 = 1, x2 = 2, x1 = 2, x2 = −1 1  x=   2

2 x=  − 1

Application 

Stretching an elastic membrane 

An elastic membrane in the x1x2-plane with boundary circle x12+x22=1 is stretched so that point P:(x1,x2) goes over into point Q:(y1,y2) given by

y1 = 5 x1 + 3 x2  y1  5 3  x1  = Ax = principal ; in components , directions of (1) y =  Find     the directions, that is, the y 2 = 3 x1 of + 5the x2 5  x2x of P for which the direction 3 vector  y2 the position position vector y of Q is the same or exactly opposite. What shape does the boundary circle take under this deformation.

Solution 

We are looking for vectors x such that y=λ x. Since y=Ax, this give Ax= λ x, an equation of the form (1), an eigenvalue problem. 0 = ( 5 − λ ) x1 + 3 x2 Ax = λx → ( A − λI ) x = 0 → 0 = 3 x1 + ( 5 − λ ) x2

5−λ 3 2 = ( 5 − λ ) − 9 = 0, λ1 = 8, λ2 = 2 3 5−λ

Solution… 

Using λ

1

− 3 x1 + 3 x2 = 0 x2 = x1, arbitrary 1 → → x =  , 3 x1 − 3 x2 = 0 x1 = 1, x2 = 1 1 45o direction 3 x1 + 3Using x2 = 0 λ 2 x2 = − x1, arbitrary 1 → → x =  , 3 x1 + 3x2 = 0 x1 = 1, x2 = −1 − 1 135o direction

Leontief Input-Output Model in Economics (Gareth, 1993 page 96) 

Wassily Leontief received a Nobel Prize in 1973 for introducing input-output model that is used to analyze the interdependence of economies.   

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Consider n interdependent industries. Let aij = amount of commodity i in $1 of commodity j. Let di = demand of open sector (consumer and government) from industry i. xi = total output of industry i necessary to meet the demand of all n industry and open sector.

Leontief Input-Output Model in Economics (Gareth, 1993 page 96)… 

Equations:   

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X = AX + D (X-AX)=D (I-A)X=D

D is thus the gross national products GNP of the economy.

Example 

An economy consisting of three industries having the following input-output matrix A. Determine the output levels required of the industries to meet the demands of the other industries and of the open sector in each case. 1 1 3  5 1 A= 2 0 

5 10   9  6 12   1 0 , D = 12, 9, 18 , 2  16 8 32 1 0 5 

Symmetric, SkewSymmetric, and Orthogonal Matrices

Definitions 

A real square matrix A=[ajk ] is called 

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symmetric if transposition leaves it unchanged, AT=A, skew-symmetric if transposition gives the negative of A, AT=-A, Orthogonal if transpostion gives the inverse of A, AT=A-1 .

Any real square matrix A may be written as sum of a symmetric matrix R and a skewsymmetric S, where R=1/2(A+AT) and S=1/2(A-AT)

Theorem 

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The eigenvalues of a symmetric matrix are real. The eigenvalues of skewsymmetric matrix are pure imaginary or zero.

Theorem 

An orthogonal transformation preserves the value of the inner T product of vectors a•b = a b (a and b are column vectors). Hence, it preserves also the length or norm of a vector in Rn given by

a = a•a = a a T

Theorem 

A real square matrix is orthogonal if and only if its column avectors 1 ,  , an (and also its row vectors) from an orthonormal system, that is,

0 if j ≠ k a j • ak = a ak  1 if j = k T j

Theorem 

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The determinant of an orthogonal matrix has the value of +1 or –1. The eigenvalues of an orthogonal matrix A are real or complex conjugates in pairs and have absolute value of 1.

Complex Matrices Hermitian, SkewHermitian, Unitary

Definitions  

The conjugate of matrix A is .A A square matrix A=[akj ] is called A T = A, that is, a jk = a jk

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Hermitian if

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T A = − A, that is, a jk = − a jk Skew-Hermitian if

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Unitary if

A T = A −1