256 Math Applied Honors Calculus IV: Differential Equations Fall 2006 Homework Set 11 Due Wednesday, December 13
Reading: Review sections 7.1, 7.2, and 7.3. Read sections 7.4, 7.5, 7.6, and 7.7 Problems to Study: Section 7.2, #1, 2, 4, 6, and 8. Practice problems with matrix algebra: addition, multiplication, transpose, adjoint, dot and inner products. Section 7.2, #10, 12, and 18. verses of matrices. Practice nding in-
Section 7.1, #22. A mixing problem. Section 7.3, #12. Linear independence of vectors of functions. Section 7.3, #21. Finding the eigenvalues and eigenvectors of a matrix. Eigenvalues and Matrix Algebra. Consider the matrices A= 1 2 0 1 and B= 2 1 1 0 .
Section 7.5, #914. Practicing solving constantcoefcient linear homogeneous systems. Eigenvalues and eigenvectors. Problems to Hand In: Section 7.1, #3. Writing higher-order differential equations and initial-value problems as rst-order systems. Section 7.1, #15 and 16. Properties of linear 2 2 rst-order systems. After you have worked these two problems, notice that the equation in number 15 can be written in compact matrix form as dx = P(t)x dt where P(t) is the matrix of coefcients and x is the column vector (x, y)T , and similarly that the in equation number 16 can be written compactly as dx = P(t)x + g(t) dt where g(t) is the column vector of forcing functions. In this notation, there is no particular reference to the number of equations or unknowns, in other words, x and g(t) could be n-component vectors and P(t) could be an n n matrix. Now repeat problems 15 and 16 in this general setting, using matrix notation (say, taking x(1) and x(2) to be the two solutions in each case). Therefore the two properties established for 22 rst-order systems actually hold for n n systems. 1
Find the eigenvalues (no need to nd the eigenvectors) of A, B, AB, and A+B. From your results, explain whether in general you can expect an eigenvalue of a product of two matrices to be a ...