91影视

Prove the part of Theorem 7.2.8 that concerns the trace: If an n 脳 n matrix A has n eigenvalues 位₁, . . . , 位_n, listed with their algebraic multiplicities, then tr A = 位₁+路 路 路+位_n.

22: Consider an arbitrary n 脳 n matrix A. What is the relationship between the characteristic polynomials of A and AT ? What does your answer tell you about the eigenvalues of A and AT ?

(a). If 2i is an eigenvalue of a real 2 脳 2 matrix A, find${\mathit{A}}^{\mathbf{2}}$.

(b). Give an example of a real 2 脳 2 matrix A such that all the entries of A are nonzero and 2i is an eigenvalue of A. Compute${\mathit{A}}^{\mathbf{2}}$and check that your answer agrees with part (a).

Consider an upper triangular $\mathbf{n}\mathbf{脳}\mathbf{n}$matrix Awith${\mathbf{a}}_{\mathbf{ij}}\mathbf{鈮�}\mathbf{0}$for$\mathbf{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{鈥�}\mathbf{,}\mathbf{m}$and${\mathbf{a}}_{\mathbf{ij}}\mathbf{=}\mathbf{0}$for$\mathbf{i}\mathbf{=}\mathbf{m}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{鈥�}\mathbf{n}$. Find the algebraic multiplicity of the eigenvalueof. Without using Theorem 7.3.6, what can you say about the geometric multiplicity?

In an unfortunate accident involving an Austrian truck, 100kgof a highly toxic substance are spilled into Lake Sils, in the Swiss Engadine Valley. The river Inn carries the pollutant down to Lake Silvaplana and later to Lake St. Moritz.

This sorry state, tweeks after the accident, can be described by the vector

$\begin{array}{r}\stackrel{鈫�}{x}\left(t\right)=\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\end{array}\right]\begin{array}{c}\mathrm{pollutant}\mathrm{in}\mathrm{lake}\mathrm{sils}\\ \mathrm{pollutant}\mathrm{in}\mathrm{lake}\mathrm{silvaplana}\\ \mathrm{pollutant}\mathrm{in}\mathrm{lake}\mathrm{st}.\mathrm{Mortiz}\end{array}\\ \end{array}\}\left(\mathrm{in}\mathrm{kg}\right)$

Suppose that$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathbf{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left[\begin{array}{ccc}0.7& 0& 0\\ 0.10& 6& 0\\ 00& 20& 8\end{array}\right]\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$

1. Explain the significance of the entries of the transformation matrix in practical terms.
2. Find closed formulas for the amount of pollutant in each of the three lakesweeks after the accident. Graph the three functions against time (on the same

axes). When does the pollution in Lake Silvaplana reach a maximum

Find a $\mathbf{2}\mathbf{脳}\mathbf{2}$matrix A such that $\left[\begin{array}{c}3\\ 1\end{array}\right]$and $\left[\begin{array}{c}1\\ 2\end{array}\right]$ are eigenvectors of A , with eigenvalues 5 and 10 , respectively.

Consider a rotation$\mathbf{T}\mathbf{}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{3}}$in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vector$\mathbf{T}\mathbf{}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{in}\mathbf{}{\mathbf{鈩�}}^{\mathbf{3}}$with$\mathbf{T}\mathbf{}\left(\stackrel{鈫�}{v}\right)\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" ${\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{位}\mathbf{\right)}$. Pay attention to the intercepts with both axes. Use Theorem 7.1.4.

Find a basis of the linear space Vof all$\mathbf{2}\mathbf{脳}\mathbf{2}$matrices Afor which$\left[\begin{array}{c}1\\ -3\end{array}\right]$ is an eigenvector, and thus determine the dimension of V.

If $\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$is any nonzero vector in ${\mathit{R}}^{\mathbf{2}}$ , what is the dimension of the space Vof all $\mathbf{2}\mathbf{脳}\mathbf{2}$matrices for which $\stackrel{\mathbf{鈬赌}}{\mathbf{v}}$is an eigenvector?

If a matrix A has k distinct eigenvalues, then $\left(A\right)\mathbf{鈮�}\mathbf{K}$