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linear algebra

Questions and Answers of Linear Algebra

Convert the second order coupled system of ordinary differential equations into a first order system involving four variables.
Solve the second order coupled system of ordinary differential equations by converting it into a first order system involving four variables.
(b) Prove that they are linearly independent.
(a) Explain how to solve the inhomogeneous ordinary differential equationdu/dt = Au + b dtwhen b is a constant vector belonging to mg A. Look at v(t) = u(t) - u* where u* is an equilibrium
True or false: The phase plane trajectories (9.10) for (c1, c2)1 ≠ 0 are hyperbolas.
(b) Explain why the two systems have the same phase portraits, but the direction of motion along the trajectories is reversed.(c) Apply time reversal to the system(s) you derived in Exercise
(b) How are the solution trajectories of the two systems related?
Let A be a constant n x n matrix, and suppose u(t) is a solution to the system(a) Show that all derivatives dku/dtk k = 1, 2....... are also solutions. (b) Show that
Each solution to a phase plane system moves at a constant speed along its trajectory.
Use a three-dimensional graphics package to plot solution curves (t, μ1(t), μ2(t))T of the phase plane systems in Exercise 9.1.1. Discuss their shape and explain how they are related to the phase
Classify the following systems according to whether the origin is(i) Asymptotically stable,(ii) Stable, or(iii) Unstable(a)(b) (c) (d) (e) (f) (g) (h)
Let A be a real 5 x 5 matrix, and assume that A has eigenvalues i, - i, -2, - 1 (and no others). Is the zero solution to the linear system u = A u necessarily stable? Explain. Does your answer change
Let u(t) solve Let v(r) = u(-t) be its time reversal.(a) Write down the linear system satisfied by v(t). Then classify the following statements as true or false. Explain your answers.(b) If is
Let u(t) solve Let v(r) = u(-t) be its time reversal.(a) Write down the linear system satisfied by v(t). Then classify the following statements as true or false. Explain your answers.(b) If is
If A is a symmetric matrix, then the system has an asymptotically stable equilibrium solution.
True or false:
(a) Find all equilibrium solutions.(b) Prove that all non-constant solutions decay exponentially fast to some equilibrium. What is the decay rate?(c) Is the origin(i) Stable,(ii) Asymptotically
(a) Let H(u, v) = au2 + buv + cv2 be a quadratic function. Prove that the non-equilibrium trajectories of the associated Hamiltonian system and those of the gradient flow are mutually orthogonal,
Suppose that u(t) satisfies the gradient flow system (9.21). (a) Prove that d/dt q(u) = ||ku||2. (b) Explain why if u(t) is any nonconstant solution to the gradient flow, then q(u(t)) is a strictly
Write out the formula for the general real solution to the system in Example 9.16 and verify its stability.
The law of conservation of energy states that the energy in a Hamiltonian system is constant on solutions. (a) Prove that if u(t) satisfies the Hamiltonian system (9.22), then H(u(t)) ≡ c is a
Prove Lemma 9.14.Let x, v be real and k > 0. A function of the formf(t) = tkeut cos v t or tk eutsin v t (9.17)
Prove Proposition 9.17.A particularly important class of systems are the linear gradient flows in which AT is a symmetric, positive definite matrix. According to Theorem 8.23, all the eigenvalues of
Write out and solve the gradient flow system corresponding to the following quadratic forms:(a) μ2 + v2(b) m v(c) 4μ2 - 2m v + v2(d) 2u2 - u v - 2u w + 2v2 - vw + 2w2
Write out and solve the Hamiltonian systems corresponding to the first three quadratic forms in Exercise 9.2.3. Which of them are stable?
Which of the following 2 x 2 systems are gradient flows? Which are Hamiltonian systems? In each case, discuss the stability of the zero solution.(a)(b) (c) (d) (e)
A nonzero linear 2 x 2 gradient flow cannot be a Hamiltonian flow.
(a) Show that the matrixhas Î» = ± i as incomplete complex conjugate eigenvalues. (c) Explain the behavior of a typical solution. Why is the zero solution not stable?
Let A be a real 3 x 3 matrix, and assume that the linear system has a periodic solution of period P. Prove that every periodic solution of the system has period P. What other types of solutions can
Are the conclusions of Exercise 9.2.8 valid when A is a 4 x 4 matrix?
For each the following:(b) Find the eigenvalues and eigenvectors of A. (c) Find the general real solution of the system. (d) Draw the phase portrait, indicating its type and stability
For each of the following systems(i)(ii) (iii) (a) Find the general real solution. (b) Using the solution formulas obtained in part (a), plot several trajectories of each system. On your graphs,
Classify the following systems, and sketch their phase portraits.(a)(b) (c) (d)
Sketch the phase portrait for the following systems:(a)(b) (c) (d) (e)
(a) Solve the initial value problem(b) Sketch a picture of your solution curve u(t), indicating the direction of motion. (c) Is the origin O (i) Stable? (ii) Asymptotically stable? (iii)
Justify the solution formulas (9.31) and (9.32).
Find the exponentials e'A of the following 2x2 matrices:(a)(b) (c) (d) (e) (f)
Explain in detail why the columns of etA form a basis for the solution space to the system u = A u.
Prove formula (9.43). Fix s and prove that, as functions of t, both sides of the equation define matrix solutions with the same initial conditions. Then use uniqueness.
Prove formula (9.43). Fix s and prove that, as functions of t, both sides of the equation define matrix solutions with the same initial conditions. Then use uniqueness.
Prove that A commutes with its exponential: A etA = etAA. Prove that both are matrix solutions to 0 = AU with the same initial conditions.
Prove that et(λ - λt) ≡ e-tλ etA by showing that both sides are matrix solutions to the same initial value problem.
(a) Prove that the exponential of the transpose of a matrix is the transpose of its exponential: etAT = (etA)T.
Prove that if A = S B S-l are similar matrices, then so are their exponentials: etA = SetB S-l.
Diagonalization provides an alternative method for computing the exponential of a complete matrix. (a) First show that if D = diag(d1,... ,dn) is a diagonal matrix, so is etD =
Justify the matrix Leibniz rule (9.40) using the formula for matrix multiplication.
Let A be a real matrix. (a) Explain why eA is a real matrix. (b) Prove that det eA > 0.
Determine the matrix exponential e'A for the following matrices:(a)(b) (c) (d)
Show that tr A = 0 if and only if det etA = I for all t.
Prove that if λ is an eigenvalue of A, then etλ is an eigenvalue of etA. What is the eigenvector?
Show that the origin is an asymptotically stable equilibrium solution to if and only if
Let A be a real square matrix and eA its exponential. Under what conditions does the linear system have an asymptotically stable equilibrium solution?
Prove that if U(t) is any matrix solution to
(a) Show that U(t) satisfies the matrix differential equation if and only if U(t) = CetB, where C = U(0).(b) Show that if U(0) is nonsingular, then U(t) also satisfies a matrix differential equation
dnu/dtnis a linear combination of
Prove that if is a block diagonal matrix, then so is
(a) Prove that if J0.n is an n x n Jordan block matrix with 0 diagonal entries, cf. (8.49), then(b) Determine the exponential of a general Jordan block matrix JÎ»,n Use Exercise
Prove that if λ is an eigenvalue of A with multiplicity k, then etA is an eigenvalue of etA with the same multiplicity. Combine the Jordan canonical form (8.51) with Exercises 9.4.17. 28.
Verify the determinant formula of Lemma 9.28 for the matrices in Exercises 9.4.1 and 9.4.2.In Exercises 9.4.1Find the exponentials e'A of the following 2x2 matrices:(a)(b) (c) (d) (e) (f) In
By a (natural) logarithm of a matrix B we mean a matrix A such that eA ‰¡ B.(a) Explain why only nonsingular matrices can have a logarithm.(b) Comparing Exercises 9.4.6-8, explain why the matrix
Solve the following initial value problems:(a)(b) (c) (d) (e)
Solve the following initial value problems:(a)(b)
Suppose that Î» is not an eigenvalue of A. Show that the inhomogeneous system has a solution of the form u*(r) = eÎ»t w, where w is a constant vector. What is the general
(a) Write down an integral formula for the solution to the initial value problem du/dt = Au + b. u(0) = 0. dt where b is a constant vector. (b) Suppose b ∈ mg A. Do you recover the solution you
Find the one-parameter groups generated by the following matrices and interpret geometrically: What are points? The trajectories? What are the fixed(a)(b)(c)(d)(e)
Write down the one-parameter groups generated by the following matrices and interpret. What are the trajectories? What are the fixed points?(a)(b) (c) (d) (e)
(a) Find the one-parameter group of rotations generated by the skew-symmetric matrix(b) As noted above, etA represents a family of rotations around a fixed axis in R3. What is the axis?
Let 0 ≠ v ∈ R3. (a) Show that the cross product Lv[x] = v x x defines a linear transformation on R3. (b) Find the 3 x 3 matrix representative Av of L, and show that it is skew-symmetric. (c) Show
Find eA when A =(a)(b) (c) (d) (e)
Let
Given a unit vector ||u|| = 1 in R3, let A = Au be the corresponding skew-symmetric 3 x 3 matrix that satisfies A x = u x x, as in Exercise 9.4.39. (a) Prove the Euler-Rodrigues formula etA = I +
Choose two of the groups in Exercise 9.4.36 or 9.4.37, and determine whether or not they commute by looking at their infinitesimal generators. Then verify your conclusion by directly computing the
(a) Prove that the commutator of two upper triangular matrices is upper triangular. (b) Prove that the commutator of two skew symmetric matrices is skew symmetric. (c) Is the commutator of two
Prove that the Jacobi identity [[A, B]. C] + ([C. A], B] -([[B, C], A] = O is valid for any three n x n matrices.
Let A be an n x n matrix whose last row has all zero entries. Prove that the last row of eTn is e"r = (0..........0. I).
Letbe in block form, where B is an n x n matrix, c ˆˆ Rn, while 0 denotes the zero row vector with n entries. Show that its matrix exponential is also in block form Can you find a formula
According to Exercise 7.3.9. any (n + 1) x (n + 1) matrix of the block formin which A is an n x n matrix and b ˆˆ Rn can be identified with the affine transformation F[x] = Ax + a on Rn.
Solve the indicated initial value problems by first exponentiating the coefficient matrix and then applying formula (9.41):(a)(b) (c)
LetShow that eA = I.
(a) Let A be a 2 x 2 matrix such that tr A = 0 and δ = √det A > 0. Prove that eA = (cos δ) I + sin δ/δ A. Use Exercise 8.2.51. (b) Establish a similar formula when det A < 0. (c) What if det A
Suppose the bottom support in the mass-spring chain in Example 9.36 is removed. (a) Do you predict that the vibration rate will (i) Speed up. (ii) Slow down, or (iii) Stay the same? (b) Verify your
(a) Describe, quantitatively and qualitatively, the normal modes of vibration for a mass-spring chain consisting of 3 unit masses, connected to top and bottom by unit springs. (b) Answer the same
Find the vibrational frequencies for a mass-spring chain with n identical masses, connected by n + 1 identical springs to both top and bottom supports. Is there any sort of limiting behavior as n →
Suppose you are given n different springs. In which order should you connect them to unit masses so that the mass-spring chain vibrates the fastest? Does your answer depend upon the relative sizes of
Suppose the illustrated planar structure has unit masses at the nodes and the bars are all of unit stiffness.(a) Write down the system of differential equations that describes the dynamical
When does a real first order linear system have a quasi-periodic solution? What is the smallest dimension in which this can occur?
Find the general solution to the following systems. Distinguish between the vibrational and unstable modes. What constraints on the initial conditions ensure that the unstable modes are not
(a) Find an orthogonal matrix Q and a diagonal matrix Î› such that K = Q Î› QT.(b) Is K positive definite?(c) Solve the second order system d2u/t2 = A u subject to the initial
Answer Exercise 9.5.17 whenIn Exercise 9.5.17 (a) Find an orthogonal matrix Q and a diagonal matrix Î› such that K = Q Î› QT. (b) Is K positive definite? (c) Solve the second
Compare the solutions to the mass-spring system (9.61) with tiny spring constant k = ε
Find the vibrational frequencies and instabilities of the following structures, assuming they have unit masses at all the nodes. Explain in detail how each normal mode moves the structure: (a) The
Discuss the three-dimensional motions of the tri-atomic molecule of Example 9.37. Are the vibrational frequencies the same as the one-dimensional model?
Assuming unit masses at the nodes, find the vibra-tional frequencies and describe the normal modes for the following planar structures. What initial conditions will not excite its instabilities
Answer Exercise 9.5.22 for the three-dimensional motions of a regular tetrahedron.
(a) Show that if a structure contains all unit masses and bars with unit stiffness, c1 = 1. Then its frequencies of vibration are the nonzero singular values of the reduced incidence matrix. (b) How
Prove that if the initial velocity satisfies = b e corng A, then the solution to the initial value problem (9.65, 71) remains bounded.
Find the general solution to the system (9.77) for the following matrix pairs:(a)(b) (c) (d) (e) (f)
A mass-spring chain of two masses, m1 = 1 and m2 = 2, connected to top and bottom supports by identical springs with unit stiffness. The upper mass is displaced by a unit distance. Find the
Answer Exercise 9.5.27 when the bottom support is removed.
Suppose you have masses m1 = 1, m2 = 2, m3 = 3, connected to top and bottom supports by identical unit springs. Does rearranging the order of the masses change the fundamental frequencies? If so,

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