18.06 Problem Set 8Due Wednesday, 22 April 2009 at 4pm in 2-106.1. If A is real-symmetric, it has real eigenvalues. What can you say about the eigenvalues if A is real andanti-symmetric (A = −AT)? Give both a general explanation for any n × n A (similar to what we didin class and in the book) and check by finding the eigenvalues a 2× 2 anti-symmetric example matrix.2. Find an orthogonal matrix Q that diagonalizes A =−2 66 7, i.e. so that QTAQ = Λ where Λ isdiagonal. What is Λ?3. Even if the real matrix A is rectangular, the block matrix B =0 AAT0is symmetric. An eigenvectorx of B satisfies Bx = λ x with:x =yz,0 AAT0yz= λyz,and thus Az = λ y and ATy = λ z.(a) Show that −λ is also an eigenvalue of B, with the eigenvector (y,−z).(b) Show that ATAz = λ2z, so that λ2is an eigenvalue of ATA.(c) Show that λ2is also an eigenvalue of AATby finding a corresponding eigenvector.(d) If A = I (2 × 2), find all four eigenvalues and eigenvectors of B.4. True or false (give a reason if true, or a counter-example if false).(a) A matrix with real eigenvalues and real eigenvectors is symmetric.(b) A matrix with real eigenvalues and orthogonal eigenvectors is symmetric.(c) The inverse of a symmetric matrix is symmetric.(d) The eigenvector matrix S of a symmetrix matrix is symmetric.(e) A complex symmetric matrix has real eigenvalues.(f) If A is symmetric, then eiAis symmetric.(g) If A is Hermitian, then eiAis Hermitian.5. For which s is A positive definite?A =s −4 −4−4 s −4−4 −4 s.16. If A has full column rank, and C is positive-definite, show that ATCA is positive definite. (Recall thatATCA is an important matrix; for example, it arose in lecture 13 on graphs and networks, section 8.2of the text.)7. For f1(x,y) = x4/4 + x2+ x2y + y2and f2(x,y) = x3+ xy − x, find the second-derivative matrices H1and H2, where:H =∂2f∂ x2∂2f∂ x∂ y∂2f∂ y∂ x∂2f∂ y2.Find the minimum point of f1(and check that H1is positive-definite there). Find the saddle point off2(look only where the first derivatives are zero, and check that H2has two eigenvalues with oppositesigns).8. (a) Give an explicit formula for uk= Aku0, where A =0 1−1 0and u0=1 2T.(b) Although you should find that A’s eigenvalues and eigenvectors are not real, give explicit valuesfor u100, u101, u102, u103, showing that your formula gives real results.(c) uk+n= ukfor what value(s) of n?9. For what (real) values of s does du/dt = Au have exponentially growing solutions, whereA =−1 s2
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