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e4-sample

# e4-sample - A is given by t 2 t 2-1 3 t-2 2(a What is the...

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Math 204. Final Exam (practice) December 5th, 2010 This is a list of practice problems for the material we covered after exam 3. You should also practice problems from previous exams since the final is cumulative. 1. Find the eigenvalues and eigenvectors for the matrix A = 1 2 2 1 . 2. Let p be a polynomial, let A C n × n , let λ be an eigenvalue for the matrix A . Prove the following: (a) p ( λ ) is an eigenvalue of p ( A ). (b) Every eigenvalue of p ( A ) is of the form p ( λ ). (c) The eigenvalues of A are non-zero if and only if A is invertible. (d) If A is invertible, then λ - 1 is an eigenvalue of A - 1 . 3. Consider the matrix A all of whose entries are equal to 1. Find the eigenvalues of A . Compute their algebraic and geometric multiplicities and find a basis for each of the eigenspaces. 4. Suppose that A R n × n is a symmetric matrix and that the characteristic polynomial of
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Unformatted text preview: A is given by t 2 ( t 2-1) 3 ( t-2) 2 . (a) What is the value of n . (b) Compute the trace and determinant of A . (c) Find the eigenvalues of A . 5. Suppose that A ∈ R n and the characteristic polynomial of A is ( t 2 + 4) 2 ( t-1) 2 . Is A symmetric? Why or why not. 6. Let T be a linear transformation on C n . Recall that the adjoint of T , denoted T * satisﬁes the equation h Tx,y i = h x,T * y i for all x,y ∈ C n . (a) If T = T * , show that the eigenvalues of T are real. (b) If TT * = T * T , then ker( T k ) = ker( T ) for all natural numbers k . (c) If T * T = TT * , then ker( T ) = ker( T * ). (d) In general show that λ is an eigenvalue of T if and only if λ is an eigenvalue of T * . 1...
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