MAT_167_WQ_2019_HW_03_WITHOUT_SOLUTIONS.pdf - MAT 167...

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MAT 167 HOMEWORK 03 WITHOUT SOLUTIONS WQ 2019 Problem 00 First of all, you might want to read the relevant parts of Sections 3.3 & 3.6, Chapter 4, and Chapter 5 in Eldén. Also you can review eigenvalues and eigenvectors in Sections 6.1 and 6.4 of Strang. Sections 6.2 and 6.5 of Strang are both important and useful, but may not be needed for this HW. Problem 01 (30 points) Let A R m × m be a symmetric matrix. You will need to know the following facts. • As you have already learned in MAT 22A or MAT 67, an eigenvector of A is a nonzero vector x C m such that A = λ x for some λ C , the corresponding eigenvalue where C denotes the complex numbers; i.e. all numbers of the form z = a + b i where a and b are real numbers and i = p - 1 is the square root of - 1. • The complex number z = a - b i is the complex conjugate of z = a + b i and if z 1 and z 2 are any two complex numbers we have z 1 z 2 = z 1 z 2 . • Finally, the inner product or dot product of two complex vectors x 1 , x 2 C is given by x 1 T x 2 so that the two norm of the vector x C is || x || 2 def = x T x In the following problem, you may assume that all of the eigenvalues of A are distinct . (a ) (15 points) Prove that all of the eigenvalues of A are real. [Hint: If λ is a (complex) eigenvalue of A with eigenvector x

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