The Use of the Power Method to Find Dominant Eigenvalues of Matrices Page: 1
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CHAPTER I
INTRODUCTION
In this chapter, we define terms and prove lemmas and theorems which are
necessary for the rest of the paper. We also see an example which demonstrates the
ideas discussed.
Given an nxn matrix, A, an eigenvalue of the matrix A is a number, A, such
that Au = Au for some non-zero vector, u. The vector u is called an eigenvector
corresponding to the eigenvalue A. An eigenvalue may also be called a proper value,
a characteristic value, or a latent root, just as an eigenvector may be called a proper
vector, a characteristic vector, or a latent vector. Note that given an eigenvalue, A,
the corresponding eigenvector u is not unique, since any multiple of u would still
be an eigenvector (for any real number k, A(ku) = k(Au) = k(Au) = A(ku)). So
direction of an eigenvector is preserved by the linear transformation represented by
A but not necessarily distance. Note that there are matrices, such as (1 j)'
which change the direction of every nonzero vector in the space. These matrices
have complex eigenvalues but no real ones. The set of all vectors u such that
Au = AU is called the eigenspace for the eigenvalue A. An eigenspace is a vector
space, so we can find a basis for any eigenspace.
A dominant eigenvalue for a matrix, A, with n eigenvalues, 1, A2,.-- , An, is
an eigenvalue, A,,, such that Vi E {1,...,n}, lAmi > lAid when Ai 0 Am. For
I
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Cavender, Terri A. The Use of the Power Method to Find Dominant Eigenvalues of Matrices, thesis, July 1992; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc501200/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .