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Chapter 5: Q7.6-23E (page 267)


For the matrix A,find real closed formulas for the trajectory x鈫�(t+1)=Ax炉(t)wherex鈫�=[01]. Draw a rough sketchA=[-0.51.5-0.61.3]

Short Answer

Expert verified

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time.

Step by step solution

01

Define Trajectory

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time.

02

Finding the Value

03

Solving the equation

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Most popular questions from this chapter

Question: A is a \({\bf{7}} \times {\bf{7}}\) matrix with three eigenvalues. One eigenspace is two-dimensional and one of the other eigenspaces is three-dimensional. Is it possible that A is not diagonalizable? Justify your answer.

Question 19: Let \(A\) be an \(n \times n\) matrix, and suppose A has \(n\) real eigenvalues, \({\lambda _1},...,{\lambda _n}\), repeated according to multiplicities, so that \(\det \left( {A - \lambda I} \right) = \left( {{\lambda _1} - \lambda } \right)\left( {{\lambda _2} - \lambda } \right) \ldots \left( {{\lambda _n} - \lambda } \right)\) . Explain why \(\det A\) is the product of the n eigenvalues of A. (This result is true for any square matrix when complex eigenvalues are considered.)

Let \(A\) be a real \(2 \times 2\) matrix with a complex eigenvalue \(\lambda = a - bi\)\(\left( {b \ne 0} \right)\) and an associated eigenvector \({\bf{v}}\) in \({\mathbb{}^2}\).

  1. Show that \(A({\mathop{\rm Re}\nolimits} {\bf{v}}) = a{\mathop{\rm Re}\nolimits} {\bf{v}} + b{\mathop{\rm Im}\nolimits} {\bf{v}}\) and \(A({\mathop{\rm Im}\nolimits} {\bf{v}}) = - b{\mathop{\rm Re}\nolimits} {\bf{v}} + a{\mathop{\rm Im}\nolimits} {\bf{v}}\). (Hint: Write \({\bf{v}} = {\mathop{\rm Re}\nolimits} {\bf{v}} + i{\mathop{\rm Im}\nolimits} {\bf{v}}\), and compute \(A{\bf{v}}\).)
  2. Verify that if \(P\) and \(C\) are given as in Theorem 9, then \(AP = PC\)

In Exercises \({\bf{3}}\) and \({\bf{4}}\), use the factorization \(A = PD{P^{ - {\bf{1}}}}\) to compute \({A^k}\) where \(k\) represents an arbitrary positive integer.

4. \(\left( {\begin{array}{*{20}{c}}{ - 2}&{12}\\{ - 1}&5\end{array}} \right) = \left( {\begin{array}{*{20}{c}}3&4\\1&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}2&0\\0&1\end{array}} \right)\left( {\begin{array}{*{20}{c}}{ - 1}&4\\1&{ - 3}\end{array}} \right)\)

Question: In Exercises 31 and 32, let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace.

31. T is the transformation on \({\mathbb{R}^2}\) that reflects points across some line through origin.
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