SrivastavaSP21ws16ComplexEigenstuff - SLC Math 54 Adjunct Faculty Instructor Location Office Hours Prof Srivastava Brittnee Mazorra

# SrivastavaSP21ws16ComplexEigenstuff - SLC Math 54...

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SLC Math 54 Adjunct Faculty: Prof. Srivastava Instructor: Brittnee Mazorra, [email protected] Location: TuTh 2-3:30pm, Zoom Office Hours: TBD in SLC Virtual Drop-In Worksheet 16: Complex Eigenstuff (5.5)) Complex Eigenstuff 1. Consider the matrix A = 0 - 1 1 0 . (a) Show A 2 = - I . (b) Suppose λ is an eigenvalue for A corresponding to x . Show that λ = ± i using the fact that A 2 = - I . (we could of course use determinants to show this as well) (c) Find the eigenvectors of A . 2. For each of the following matrices, find all eigenvalues and bases for the corresponding eigenspaces. (a) 3 - 4 4 3 (b) 1 - 2 3 1 (c) 1 - 1 - 1 1 - 1 0 1 0 - 1 3. Show that if a, b R , then the eigenvalues of A = a - b b a are a ± bi , with corresponding eigenvectors i 1 , - i 1 . 4. Find an invertible matrix P and a matrix C of the form a - b b a such that A = PCP - 1 . (a) A = 3 - 4 4 3 (b) A = 1 - 2 3 1 (c) A = - 1 - 2 1 1 (d) A = 3 - 2 1 3 5. Many properties of complex numbers carry over to complex matrix algebra. For example, if A R n × n then A x = A x . Show that if