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examples152-ch8

# examples152-ch8 - MATH 152 Fall 2006-07 Applied Linear...

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MATH 152 Fall 2006-07 Worked Examples Dr. Tony Yee Department of Mathematics The Hong Kong University of Science and Technology September 1, 2006

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Contents Table of Contents iii 1 Introduction 1 2 First-Order Differential Equations 5 3 Second-Order Linear Equations 39 4 Laplace Transform 83 5 Matrix 127 6 Systems of Linear Equations 135 7 Euclidean Vector 155 8 Eigenvalue and Eigenvector 177 9 Systems of Differential Equations 209 10 Orthogonality 231 iii

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Chapter 8 Eigenvalue and Eigenvector ¥ Example 8.1 (What is a diagonalization?) Compute the matrices A 1 , A 2 , A 3 and verify that they are all equal after multiplications. A 1 = 2 4 1 - 1 1 1 1 - 1 - 1 1 1 3 5 2 4 1 0 0 0 2 0 0 0 3 3 5 2 4 1 - 1 1 1 1 - 1 - 1 1 1 3 5 - 1 , A 2 = 2 4 - 1 1 1 1 1 - 1 1 - 1 1 3 5 2 4 2 0 0 0 1 0 0 0 3 3 5 2 4 - 1 1 1 1 1 - 1 1 - 1 1 3 5 - 1 , A 3 = 2 4 1 - 1 1 - 1 1 1 1 1 - 1 3 5 2 4 3 0 0 0 2 0 0 0 1 3 5 2 4 1 - 1 1 - 1 1 1 1 1 - 1 3 5 - 1 . Solution We compute A 1 only. By doing row operations, we have ˆ P I ˜ -→ ˆ I P - 1 ˜ , which gives P - 1 = 2 4 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 1 / 2 3 5 , where P = 2 4 1 - 1 1 1 1 - 1 - 1 1 1 3 5 . Hence, A 1 = 2 4 1 - 2 3 1 2 - 3 - 1 2 3 3 5 2 4 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 1 / 2 3 5 = 2 4 2 - 1 / 2 1 / 2 - 1 3 / 2 - 1 / 2 1 1 / 2 5 / 2 3 5 . Similarly, we can compute A 2 , A 3 and ﬁnally verify that they all give the same matrix. A 1 = A 2 = A 3 = 2 4 2 - 1 / 2 1 / 2 - 1 3 / 2 - 1 / 2 1 1 / 2 5 / 2 3 5 . 2 Remark . From the matrix point of view, the above result is not coincidental. The reason behind is called the diagonalizability of a matrix. Suppose an n × n matrix A is given. The matrix is said to be diagonalizable if there exists an invertible matrix P such that D = P - 1 AP is diagonal. In other words, PDP - 1 is said to be a diagonalization of A if there exist an invertible matrix P and a diagonal matrix D . Therefore the three given products of matrices are indeed three diﬀerent diagonalizations of a matrix A . 177

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8. Eigenvalue and Eigenvector Such a diagonalized representation of A turns out to have some nice applications in linear algebra. However, ﬁrst thing ﬁrst, we should learn an algorithm for ﬁnding such an invertible matrix P that can turn A into a diagonal matrix D provided that A is diagonalizable. The main ingredients here are eigenvalues and eigenvectors . In the following we ﬁrst give a formal deﬁnition for what are eigenvalues as well as eigenvectors of a given square matrix. Deﬁnition . Let A be any square matrix. A scalar λ is called an eigenvalue of A if there exists a nonzero (column) vector v such that Av = λ v . Any vector satisfying this relation is called an eigenvector of A belonging to the eigenvalue λ . Based on the discussion in Section 8.2 (Lecture Notes, page 165 ) and the above deﬁnition, we have the following theorem which serves as an algorithm for constructing diagonalizations for any diagonalizable matrices.
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examples152-ch8 - MATH 152 Fall 2006-07 Applied Linear...

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