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Find the characteristic polynomial of the $\mathbf{n}\mathbf{脳}\mathbf{n}$matrix

role="math" localid="1659601653126" $\mathbf{A}\mathbf{=}\mathbf{\left[}\begin{array}{cccccc}\mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{xxx}& \mathbf{0}& {\mathbf{a}}_{\mathbf{0}}\\ \mathbf{1}& \mathbf{0}& \mathbf{0}& \mathbf{xxx}& \mathbf{0}& {\mathbf{a}}_{\mathbf{1}}\\ \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{xxx}& \mathbf{0}& {\mathbf{a}}_{\mathbf{2}}\\ \mathbf{M}& \mathbf{M}& \mathbf{M}& \mathbf{M}\mathbf{}\mathbf{O}& \mathbf{M}& \mathbf{M}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\mathbf{}\mathbf{L}& \mathbf{0}& {\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{2}}& \mathbf{}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}\mathbf{}\mathbf{L}& \mathbf{1}& {\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}& \mathbf{}\end{array}\mathbf{\right]}$

Note that the i-th column of A is${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{i}\mathbf{+}\mathbf{1}}$, for i = 1, ..., n - 1while the last column has the arbitrary entries${\mathbf{a}}_{\mathbf{0}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$. See Exercise 51 for the special case n = 3.

For a regular transition matrix A, prove the following:

a. $\mathit{位}\mathbf{=}\mathbf{1}$is an eigenvalue of A with geometric multiplicity 1

b. If $\mathbf{位}$is any real eigenvalue of A, then $\mathbf{-}\mathbf{1}\mathbf{<}\mathbf{位}\mathbf{鈮�}\mathbf{1}$.

For Exercises 51 through 55, state whether the given set is a field (with the customary addition and multiplication).

53. The binary digits

find an eigenbasis for the given matrice and diagonalize:

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$

If v鈫抜s an eigenvector of A, then v鈫抦ust be in the kernel of Aor in the image ofA.

Consider $\mathbf{5}\mathbf{脳}\mathbf{5}$matrix A and a vector $\stackrel{\mathbf{鈫�}}{\mathbf{v}}$in${\mathbf{鈩�}}^{\mathbf{5}}$. Suppose the vectors $\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$are linearly independent, while ${\mathbf{A}}^{\mathbf{3}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\mathbf{a}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}\mathbf{bA}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}{\mathbf{cA}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$for some scalars a,b,c. We can take the linearly independent vectors $\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$and expand them to basis$\mathbf{I}\mathbf{=}\mathbf{(}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{4}\mathbf{,}\stackrel{\mathbf{鈫�}}{\mathbf{w}}\mathbf{5}\mathbf{)}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{鈩�}}^{\mathbf{5}}$.

1. Consider the matrix Bof the linear transformation $\mathbf{T}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$with respect to the basis $\mathfrak{J}$. Write the entries of the first three columns of B.
2. Explain why ${\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{位}\mathbf{\right)}\mathbf{=}{\mathbf{f}}_{\mathbf{B}}\mathbf{\left(}\mathbf{位}\mathbf{\right)}\mathbf{=}\mathbf{h}\mathbf{\left(}\mathbf{位}\mathbf{\right)}\mathbf{(}\mathbf{-}{\mathbf{位}}^{\mathbf{3}}\mathbf{+}{\mathbf{肠位}}^{\mathbf{2}}\mathbf{+}\mathbf{产位}\mathbf{+}\mathbf{a}\mathbf{)}$for some quadratic polynomial $\mathbf{h}\mathbf{\left(}\mathbf{位}\mathbf{\right)}$
3. Explain why ${\mathbf{f}}_{\mathbf{A}}\mathbf{=}\left(A\right)\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{0}}$. Here is ${\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}$the characteristic polynomial evaluated at A, that is if${\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{位}\mathbf{\right)}\mathbf{=}{\mathbf{c}}_{\mathbf{n}}{\mathbf{位}}^{\mathbf{n}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{c}}_{\mathbf{1}}\mathbf{位}\mathbf{+}{\mathbf{c}}_{\mathbf{0}}\mathbf{}\mathbf{then}\mathbf{}{\mathbf{f}}_{\mathbf{A}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{=}{\mathbf{c}}_{\mathbf{n}}{\mathbf{A}}^{\mathbf{n}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{c}}_{\mathbf{1}}\mathbf{A}\mathbf{+}{\mathbf{c}}_{\mathbf{0}}{\mathbf{l}}_{\mathbf{n}}$

Consider$\mathbf{n}\mathbf{脳}\mathbf{n}$matrix A and a vector$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$in${\mathbf{鈩�}}^{\mathbf{n}}\mathbf{,}{\mathbf{鈩�}}^{\mathbf{n}}$. Form the vectorslocalid="1659593339778" $\mathbf{v}\mathbf{鈨�}\mathbf{,}\mathbf{Av}\mathbf{鈨�}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\mathbf{v}\mathbf{鈨�}\mathbf{,}{\mathbf{A}}^{\mathbf{3}}\mathbf{v}\mathbf{鈨�}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}$and let ${\mathbf{A}}^{\mathbf{m}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$be the first redundant vector in the list. Then the vectorslocalid="1659593377664" $\mathbf{v}\mathbf{鈨�}\mathbf{,}\mathbf{Av}\mathbf{鈨�}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\mathbf{v}\mathbf{鈨�}\mathbf{,}{\mathbf{A}}^{\mathbf{3}}\mathbf{v}\mathbf{鈨�}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}\mathbf{v}\mathbf{鈨�}$are linearly independent. Note the$\mathbf{m}\mathbf{<}\mathbf{n}$. Since${\mathbf{A}}^{\mathbf{m}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$is redundant , we can write${\mathbf{A}}^{\mathbf{m}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}{\mathbf{a}}_{\mathbf{0}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{a}}_{\mathbf{m}\mathbf{-}\mathbf{1}}{\mathbf{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$ for some scalar${\mathbf{a}}_{\mathbf{0}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{a}}_{\mathbf{m}\mathbf{-}\mathbf{1}}$

Form the basislocalid="1659594375614" $\mathbf{l}\mathbf{=}\mathbf{(}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{A}}^{\mathbf{m}\mathbf{-}\mathbf{1}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{m}\mathbf{+}\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{n}}\mathbf{)}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{鈩�}}^{\mathbf{n}}$

(a ) Consider the matrix B of the linear transformation$\mathbf{T}\mathbf{\left(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\right)}\mathbf{=}\mathbf{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$with respect to the basis B in the block form$\mathbf{B}\mathbf{=}\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right]$, where${\mathbf{B}}_{\mathbf{11}}$is an$\mathbf{n}\mathbf{脳}\mathbf{n}$ matrix. Describe${\mathbf{B}}_{\mathbf{11}}$column by column, paying particular attention to the column. What you can say about the${\mathbf{B}}_{\mathbf{21}}$ ?.s

(b) Explain why ${\mathbf{f}}_{\mathbf{A}}\left(位\right)\mathbf{=}{\mathbf{f}}_{\mathbf{B}}\left(位\right)\mathbf{=}{\mathbf{f}}_{\mathbf{B}\mathbf{22}}\left(位\right){\mathbf{f}}_{\mathbf{B}\mathbf{11}}\left(位\right)\mathbf{=}{\left(-1\right)}^{\mathbf{m}}{\mathbf{f}}_{\mathbf{B}\mathbf{22}}\left(位\right)\left({位}^m-a_{m-1}{位}^{m-1}-.....-a_1位-a_0\right)$see Exercise 52

(c) Explain why ${\mathbf{f}}_{\mathbf{A}}\left(A\right)\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{0}}$. See Exercise

(d) Explain why${\mathbf{f}}_{\mathbf{A}}\left(A\right)\mathbf{=}\mathbf{0}$

find an eigenbasis for the given matrice and diagonalize:

$\mathbf{A}\mathbf{=}\left[\begin{array}{ccc}1& -1& 1\\ -1& 1& -1\\ 1& -1& 1\end{array}\right]$

For Exercises 51 through 55, state whether the given set is a field (with the customary addition and multiplication).

54. The rotation-scaling matrices of the form $\left[\begin{array}{c}p-q\\ \mathrm{qp}\end{array}\right]$, where p and q are real numbers

For Exercises 51 through 55, state whether the given set is a field (with the customary addition and multiplication).

55. The set H