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Chapter 5: Q7.6-19E (page 267)

For the Matrices A find real closed formulas for the trajectory x鈫�(t+1)=Ax鈫�(t)wherex鈫�(0)=[01]A=[2-332]

Short Answer

Expert verified

A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

Step by step solution

01

Define Matrix  

A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

02

Solving the Equation

03

Finding the Solution

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Most popular questions from this chapter

Consider the growth of a lilac bush. The state of this lilac bush for several years (at year鈥檚 end) is shown in the accompanying sketch. Let n(t) be the number of new branches (grown in the year t) and a(t) the number of old branches. In the sketch, the new branches are represented by shorter lines. Each old branch will grow two new branches in the following year. We assume that no branches ever die.

(a) Find the matrix A such that [nt+1at+1]=A[ntat]

(b) Verify that [11]and [2-1] are eigenvectors of A. Find the associated eigenvalues.

(c) Find closed formulas for n(t) and a(t).

For the matrix A, find real closed formulas for the trajectoryx鈫�(t+1)=Ax炉(t)where x鈫�=[01]. Draw a rough sketch

A=[15-27]

Apply the results of Exercise \({\bf{15}}\) to find the eigenvalues of the matrices \(\left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{2}}&{\bf{2}}\\{\bf{2}}&{\bf{1}}&{\bf{2}}\\{\bf{2}}&{\bf{2}}&{\bf{1}}\end{aligned}} \right)\) and \(\left( {\begin{aligned}{*{20}{c}}{\bf{7}}&{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{7}}&{\bf{3}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{7}}&{\bf{3}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{7}}&{\bf{3}}\\{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{3}}&{\bf{7}}\end{aligned}} \right)\).

Use mathematical induction to show that if \(\lambda \) is an eigenvalue of an \(n \times n\) matrix \(A\), with a corresponding eigenvector, then, for each positive integer \(m\), \({\lambda ^m}\)is an eigenvalue of \({A^m}\), with \({\rm{x}}\) a corresponding eigenvector.

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