eigenval - 10 Eigenvalues and Eigenvectors Fall 2003...

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10 Eigenvalues and Eigenvectors Fall 2003 Introduction To introduce the concepts of eigenvalues and eigenvectors , we consider first a three- dimensional space with a Cartesian coordinate system. Consider a vector from the origin O to a point P ; call this vector a . The components of a are ( a 1 , a 2 , a 3 ). Alternatively, we could say that point P has coordinates ( a 1 , a 2 , a 3 ). We can apply a linear transformation to vector a to obtain another vector z For any linear transformation, each component of z is some linear combination of the components of a This relationship can be expressed as z S a S a S a z S a S a S a z S a S a S a 1 11 1 12 2 13 3 2 21 1 22 2 23 3 3 31 1 32 2 33 3 = + + = + + = + + , , . or z S a i i j j j = = . 1 3 (1) This can be expressed more compactly by use of matrix language. We define: a z S = F H G G I K J J = F H G G I K J J = F H G G I K J J a a a z z z S S S S S S S S S 1 2 3 1 2 3 , , (2) Then z z z S S S S S S S S S a a a S a S a S a S a S a S a S a S a S a 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 F H G G I K J J = F H G G I K J J F H G G I K J J = + + + + + + F H G G I K J J , (3) or simply z = S a (4) In general, the direction of vector z is different from that of a But there may be special cases where z has the same direction as a. For example, suppose the transformation S represents a rotation of vector a about some fixed axis. If the direction of a happens to coincide with this axis, then a is not changed by this transformation, and z = a . More generally, if z has the same direction (but not necessarily the same magnitude) as a then z must be a scalar multiple of a. That is, z = λ a, where λ is a scalar In that case, Sa = λ a, (5)
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10-2 10 Eigenvalues and Eigenvectors or S S S S S S S S S a a a a a a a a a 11 12 13 21 22 23 31 32 33 1 2 3 1 2 3 1 2 3 F H G G I K J J F H G G I K J J = F H G G I K J J = F H G G I K J J λ λ λ λ . (6) In this case, the result of the transformation S applied to the vector a is another vector having the same direction as a . A vector a for which Eqs. (5) and (6) are valid is called an eigenvector of the transformation S , and the scalar λ is the corresponding eigenvalue . Finding Eigenvectors Several questions immediately arise: (1) How do we know that eigenvectors exist , for any given transformation S ? (2) If eigenvectors (and their corresponding eigenvalues) do exist, how can we find them? (3) Can there be more than one eigenvector for a given transformation S ? If so, how are the different eigenvectors and eigenvalues related? We'll now try to answer these questions. First, Eq. (6) can be combined with Eq. (3) and re-arranged as follows: S a S a S a S a S a S a S a S a S a a a a 11 1 12 2 13 3 21 1 22 2 23 3 31 1 32 2 33 3 1 2 3 + + + + + + F H G G I K J J = F H G G I K J J λ λ λ (7) Equating corresponding elements in Eq. (7) and re-arranging, we obtain the set of three scalar equations: ( ) , ( ) , ( ) . S a S a S a S a S a S a S a S a S a 1 2 3 1 2 3 1 2 3 0 0 0 - + + = + - + = + + - = λ λ λ . (8) If an eigenvector a exists, its components ( a 1 , a 2 , a 3 ) and its eigenvalue λ must satisfy Eqs. (8). This is a set of three simultaneous, linear, homogeneous equations. (I.e., every term contains an a
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eigenval - 10 Eigenvalues and Eigenvectors Fall 2003...

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