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nd homeworks are allowed, each student must write and submit their own, independent solutions. Let T be the 2-by-2 matrix T= T = (₂2) " with real coefficients. Everything below can be generalized to a general 2-by-2 matrix, to a general n-by-n atrix, and even beyond, but we want to keep computations tractable. Recall that a complex number À is called an eigenvalue of T if there exists a nonzero vector v such that Τυ = λυ. et us call o the set of all the eigenvalues of T. Note that À E o if and only if the matrix -b) - a

nd homeworks are allowed, each student must write and submit their own, independent solutions. Let T be the 2-by-2 matrix T= T = (₂2) " with real coefficients. Everything below can be generalized to a general 2-by-2 matrix, to a general n-by-n atrix, and even beyond, but we want to keep computations tractable. Recall that a complex number À is called an eigenvalue of T if there exists a nonzero vector v such that Τυ = λυ. et us call o the set of all the eigenvalues of T. Note that À E o if and only if the matrix -b) - a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Show your reasoning for each exercise. While discussions among students concerning the course material and homeworks are allowed, each student must write and submit their own, independent solutions. Let T be the 2-by-2 matrix with real coefficients. Everything below can be generalized to a general 2-by-2 matrix, to a general n-by-n matrix, and even beyond, but we want to keep computations tractable. Recall that a complex number A is called an eigenvalue of T if there exists a nonzero vector v such that Τυ = λυ. Let us call o the set of all the eigenvalues of T. Note that X E o if and only if the matrix -b λ-a 0 X-d, is not invertible. This last condition is in turn equivalent to 0 = det(X-T). 1. Compute det (C – T) as a function of and list all possibilities for the number of (complex) zeros this function has i.e. eigenvalues that the matrix T has. Show that this inverse has matrix form by computing the product X-T= Hint: Recall our discussion on the multiplicity of zeros, or compute explicitly the determinant and consider the cases generated by different values of the parameters a and d. 2. If Ço, then the matrix ( - T is invertible and we set R(S) := (CT)-¹. T = = (82) R(C) det (C-T) 0 = s=ºa) 3. Note that the matrix entries of R(C) are holomorphic functions of C on C\o. Describe the singularities of the top-left matrix element. in the generic case a d and ad # 0. C-d (B) det (C-T) det (C-T) 0 det (5-7) det (C-T) S-d det (CT) Warning: Like the number of zeros in Problem 1, the singularities may change type as the parameters a, b and d change. 4. Let y be a counter-clockwise circle centered at the origin that is large enough to circle all of o. Adopting the convention that a contour integral of a matrix-valued function is the matrix built from the contour integrals of the respective entries, compute - a 0 1 2πi fc²R (C) a$ 1
Transcribed Image Text:Show your reasoning for each exercise. While discussions among students concerning the course material and homeworks are allowed, each student must write and submit their own, independent solutions. Let T be the 2-by-2 matrix with real coefficients. Everything below can be generalized to a general 2-by-2 matrix, to a general n-by-n matrix, and even beyond, but we want to keep computations tractable. Recall that a complex number A is called an eigenvalue of T if there exists a nonzero vector v such that Τυ = λυ. Let us call o the set of all the eigenvalues of T. Note that X E o if and only if the matrix -b λ-a 0 X-d, is not invertible. This last condition is in turn equivalent to 0 = det(X-T). 1. Compute det (C – T) as a function of and list all possibilities for the number of (complex) zeros this function has i.e. eigenvalues that the matrix T has. Show that this inverse has matrix form by computing the product X-T= Hint: Recall our discussion on the multiplicity of zeros, or compute explicitly the determinant and consider the cases generated by different values of the parameters a and d. 2. If Ço, then the matrix ( - T is invertible and we set R(S) := (CT)-¹. T = = (82) R(C) det (C-T) 0 = s=ºa) 3. Note that the matrix entries of R(C) are holomorphic functions of C on C\o. Describe the singularities of the top-left matrix element. in the generic case a d and ad # 0. C-d (B) det (C-T) det (C-T) 0 det (5-7) det (C-T) S-d det (CT) Warning: Like the number of zeros in Problem 1, the singularities may change type as the parameters a, b and d change. 4. Let y be a counter-clockwise circle centered at the origin that is large enough to circle all of o. Adopting the convention that a contour integral of a matrix-valued function is the matrix built from the contour integrals of the respective entries, compute - a 0 1 2πi fc²R (C) a$ 1
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