notebook04.pdf - MATH2210 Notebook 4 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009\u20132018 by Jenny A Baglivo All Rights Reserved 4

# notebook04.pdf - MATH2210 Notebook 4 Spring 2018 prepared...

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MATH2210 Notebook 4 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009–2018 by Jenny A. Baglivo. All Rights Reserved. 4 MATH2210 Notebook 4 3 4.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.1.1 Definitions and Geometric Interpretations . . . . . . . . . . . . . . . . . . . . . . 3 4.1.2 Eigenspaces, Characteristic Polynomials, Characteristic Equations . . . . . . . . 5 4.1.3 Eigenanalysis and Powers; Eigenvector Bases; Special Cases . . . . . . . . . . . . 10 4.1.4 Algebraic Multiplicity and Geometric Multiplicity . . . . . . . . . . . . . . . . . 15 4.1.5 Multiplicity and Finding Eigenvector Bases . . . . . . . . . . . . . . . . . . . . . 16 4.1.6 Similar Matrices and Diagonalizable Matrices . . . . . . . . . . . . . . . . . . . . 17 4.1.7 Applications: Population Projections and Stochastic Matrices . . . . . . . . . . . 19 4.2 Orthogonality and Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.1 Inner Product, Length and Distance . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2.2 Properties of Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.3 Orthogonal Vectors, Orthogonal Sets and Orthogonal Complement . . . . . . . . 27 4.2.4 Fundamental Theorem of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . 30 4.2.5 Orthogonal Spanning Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.6 Angles, Inner Products, and Orthogonal Projections . . . . . . . . . . . . . . . . 33 4.2.7 Gram-Schmidt Orthogonalization Process . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Least Squares Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3.1 Best Approximate Solutions and the Normal Equations . . . . . . . . . . . . . . 39 4.3.2 Application: Least Squares Analyses of Data . . . . . . . . . . . . . . . . . . . . 42 4.3.3 Footnote: Eigenvalues, Eigenvectors and Least Squares Analysis . . . . . . . . . 47 1
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4 MATH2210 Notebook 4 This notebook is concerned with further matrix concepts and their applications. In particular, we will study eigenvalues, eigenvectors, orthogonality and least squares. The notes correspond to material in Chapters 5 and 6 of the Lay textbook. 4.1 Eigenvalues and Eigenvectors An eigenvalue is an “exceptional value” and an eigenvector is an “exceptional vector.” The prefix “eigen” comes from the German language meaning “owned by” or “peculiar to.” Applications of eigenvalues and eigenvectors first appeared in the literature in the 18 th century, although the prefix “eigen” was not used until the early part of the 20 th century by the mathematician David Hilbert. 4.1.1 Definitions and Geometric Interpretations Let A be a square matrix of order n , and λ (“lambda”) be a scalar. 1. Eigenvalue of a Square Matrix: The scalar λ is said to be an eigenvalue of A if A x = λ x for some nonzero vector x . 2. Eigenvector of a Square Matrix: If x 6 = O satisfies A x = λ x , then x is an eigenvector of A with eigenvalue λ . Note on Eigenvectors and Spans: If x is a nonzero eigenvector, then so is c x for each nonzero c R since A ( c x ) = c ( A x ) = c ( λ x ) = λ ( c x ) . Thus, the nonzero vectors in Span { x } are all eigenvectors of A with eigenvalue λ . Note on Geometric Interpretation: If x R n is eigenvector of A with eigenvalue λ R , and if T : R n R n is the linear transformation whose standard matrix is A , then T maps the span of x into itself: T (Span { x } ) Span { x } . Further, each vector is re-scaled by the eigenvalue λ . 3
Example 1 . Let A = 0 . 5 0 0 1 . 5 . Then 1. λ 1 = 0 . 5 is an eigenvalue of A , with corre- sponding eigenvector e 1 .