Math 1b Practical — Problem Set 7Due 3:00pm, Monday, February 28, 20111. (15 points) Find a 3 × 3 matrix with the following eigenvalues and corresp ondingeigenvectors.λ1=3, e1=⎛⎝101⎞⎠,λ2=3, e2=⎛⎝2−10⎞⎠,λ3=2, e3=⎛⎝011⎞⎠.2. (15 p oints) Supp ose that A is diagonalizable and that all eigenvalues of A have absolutevalue less than 1. Explain why An→ O as n →∞.3. (10 points) Explain why the eigenvalues of a square matrix A are the s ame as those ofA.4. (20 points) Given an m by n matrix A =(aij), the numbersnj=1aij,i=1, 2,...,m,are called the row sums of A. Now assume A is square and that the entries of A arenonnegative.(i) Give an example of a 2 × 2 nonnegative matrix B with both row sums equal to 2so that one eigenvalue of B is −2.(ii) Prove that if μ is any eigenvalue of A,then|μ| cannot exceed the maximum of therow sums of A.Suggestion: That μ is an eigenvalue of A means that for s ome nonzero column vectorx =(x1,...,xn),μxi=nj=1aijxjfor i =1, 2,...,n.Take the absolute value of b oth sides and use the tr iangle inequality to get an inequality|μ|·|xi|≤....Thenpicki0so that |xi0| is the m aximum of the values of |xi|, i =1, 2,...,n,and u s e the inequality |xi|≤|xi0| for each i.(iii) Prove that if μ is any eigenvalue of A,then|μ| cannot exceed the maximum ofthe column sums of A. Suggestion: think about A.5. (20 points) LetA1=⎛⎜⎜⎜⎜⎜⎜⎝λc2c3... cn0 λ 0 ··· 000λ ··· 000 0··· 0...............00 0··· λ⎞⎟⎟⎟⎟⎟⎟⎠,A2=⎛⎜⎜⎜⎜⎜⎜⎝λ 00... 0c2λ 0 ··· 0c30 λ ··· 0c400··· 0...............cn00··· λ⎞⎟⎟⎟⎟⎟⎟⎠,be square matrices where λ and c2,c3,...,cnare positive numbers.(i) Sketch the digraphs of A1and A2.(ii) What are the algebraic and geometric multiplicities of λ as an eigenvalue of A1and A2?(iii) What is the maximum number of linearly independent nonnegative eigenvectorsthat can be found for
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