nt4564vectr3 - Eigenvalues and Eigenvectors of Matrices Let...

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Eigenvalues and Eigenvectors of Matrices Eigenvalues and Eigenvectors of Matrices Let A be an n × n matrix with either real or complex entries a ij . Then for vectors X in R n or E n , as appropriate, the mapping from X to A X is linear, i.e., A ( α X + β Y ) = α A X + β A Y . As a consequence of this linearity property one can see that a general line, V , through the origin, which consists of all vectors of the form α V, α real, for some non-zero “generating” vector V , is carried into another line through the origin, namely A V = n α A V α real o . We say that a scalar λ is an eigenvalue of the matrix A if there is a non-zero vector Φ, in R n or E n as appropriate, such that A Φ = λ Φ. Geo- metrically this means that A carries the line Φ into itself, each vector X on that line going to the image vector λ X . Further, any non-zero vector ˜ Φ such that A ˜ Φ = λ ˜ Φ is an eigenvector of A corresponding to the eigenvalue λ . Clearly eigenvectors are not unique; in fact, any non-zero vector on the line Φ is any eigenvector of A corresponding to λ . It is also possible that there might be other vectors Ψ not on the line Φ which are eigenvectors of A corresponding to λ ; the largest number, m , of linearly independent vec- tors Φ 1 , Φ 2 , ..., Φ m qualifying as eigenvectors of A corresponding to the single eigenvalue λ is called the geometric multiplicity of the eigenvalue λ . If m = 1 the eigenvalue λ is simple , or has single multiplicity . We can write the eigenvalue-eigenvector equation (the resolvent equation ) as ( λ I - A ) Φ = 0 , Φ 6 = 0 , where I denotes the n × n identity matrix. A standard theorem of linear algebra states that such a linear homogeneous system of equations has a non- 1
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zero vector solution Φ if and only if the determinant of the matrix λ I - A is zero. That determinant turns out to be an n -th degree polynomial in λ of the form p ( λ ) = λ n + p 1 λ n - 1 + · · · + p n - 1 λ + p n . The eigenvalues of A are those numbers λ for which p ( λ ) = 0. In general, from the fundamental theorem of algebra , there are n of these. However, since the polynomial p ( λ ) may have repeated roots, so that p ( λ ) = ( λ - λ 1 ) μ 1 ( λ - λ 2 ) μ 2 · · · ( λ - λ ν ) μ ν for positive integer multiplicities μ 1 , μ 2 , · · · μ ν whose sum is n , the eigenvalues need not consist of n distinct numbers. The μ k shown here are the algebraic multiplicities of the eigenvalues λ k . The geometric and algebraic multiplicities need not be the same but we always have μ k m k .
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