LAST TOPICS

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LAST TOPICS

• Homogeneous linear equations
• Eigenvalues and eigenvectors

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Eigenvalues and Eigenvectors
An nn matrix A multiplied by n1 vector v
results in another n1 vector yAv. Thus A can be
considered as a transformation matrix. In
general, a matrix acts on a vector by changing
both its magnitude and its direction. However, a
matrix may act on certain vectors by changing
only their magnitude, and leaving their direction
unchanged (or possibly reversing it). These
vectors are the eigenvectors of the matrix. A
matrix acts on an eigenvector by multiplying its
magnitude by a factor, which is positive if its
direction is unchanged and negative if its
direction is reversed. This factor is the
eigenvalue associated with that
eigenvector. Matrix expands or shrinks any
vector lying in direction of eigenvector by
scalar factor Expansion or contraction factor is
given by corresponding eigenvalue
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HOW TO FIND THEM
Let x be an eigenvector of the matrix M. Then
there must exist an eigenvalue ? such that Mx
?x or, equivalently, Mx - ?x 0 or (M
?I)x 0 If we define a new matrix B M ?I,
then Bx 0 If B has an inverse then x
B-10 0. But an eigenvector cannot be
zero. Thus, it follows that x will be an
eigenvector of M if and only if B does not have
an inverse, or equivalently det(B)0, or det(M
?I) 0 The roots of this equation determine
the eigenvalues of M.
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