Commit 4c31b009 by Jim Hefferon

 ... @@ -29935,49 +29935,6 @@ octave:6> gplot z ... @@ -29935,49 +29935,6 @@ octave:6> gplot z \end{ans} \end{ans} \begin{ans}{Five.II.3.24} \begin{ans}{Five.II.3.24} \begin{exparts} \item The characteristic polynomial factors as $x^2-20x+75=(x-5)(x-15)$, so the eigenvalues are $\lambda_1=5$ and~$\lambda_2=15$. These are the associated eigenspaces. \begin{equation*} V_5=\set{k\colvec{1 \\ 2}\suchthat k\in\C} \quad V_{15}=\set{k\colvec{-2 \\ 1}\suchthat k\in\C} \end{equation*} For each eigenvalue, both the algebraic and geometric multiplicities are~$1$. \item The characteristic polynomial $x^3 - 6x^2 + 32$ factors into $(x+2)(x-4)^2$. The eigenvectors are $\lambda_1=-2$ and $\lambda_2=4$. Here are the associated eigenspaces. \begin{equation*} V_{-2}=\set{k\colvec{1 \\ 1 \\ 2}\suchthat k\in\C} \quad V_4=\set{k_1\colvec{1 \\ 1 \\ 0} +k_2\colvec{1 \\ 0 \\ -1} \suchthat k_1,k_2\in\C} \end{equation*} For $\lambda_1=-2$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=4$ the algebraic and geometric multiplicities are both~$2$. \item The characteristic polynomial $x^3 - 5x^2 + 8x - 4$ factors into $(x-1)(x-2)^2$. The eigenvalues are $\lambda_1=1$ and $\lambda_2=2$. Here are the associated eigenspaces. \begin{equation*} V_1=\set{k\colvec{-6 \\ 3 \\ 1}\suchthat k\in \C} \quad V_2=\set{k\colvec{1 \\ 0 \\ 0}\suchthat k\in\C} \end{equation*} For $\lambda_1=1$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=2$ the algebraic multiplicity is~$2$ but the geometric multiplicity is~$1$. \end{exparts} \end{ans} \begin{ans}{Five.II.3.25} The characteristic equation The characteristic equation \begin{equation*} \begin{equation*} 0= 0= ... @@ -30045,7 +30002,7 @@ octave:6> gplot z ... @@ -30045,7 +30002,7 @@ octave:6> gplot z \end{equation*} \end{equation*} \end{ans} \end{ans} \begin{ans}{Five.II.3.26} \begin{ans}{Five.II.3.25} The characteristic equation is The characteristic equation is \begin{equation*} \begin{equation*} 0= 0= ... @@ -30083,7 +30040,7 @@ octave:6> gplot z ... @@ -30083,7 +30040,7 @@ octave:6> gplot z \end{equation*} \end{equation*} \end{ans} \end{ans} \begin{ans}{Five.II.3.27} \begin{ans}{Five.II.3.26} \begin{exparts} \begin{exparts} \partsitem The characteristic equation is \partsitem The characteristic equation is \begin{equation*} \begin{equation*} ... @@ -30285,6 +30242,49 @@ octave:6> gplot z ... @@ -30285,6 +30242,49 @@ octave:6> gplot z \end{equation*} \end{equation*} \end{exparts} \end{exparts} \end{ans} \begin{ans}{Five.II.3.27} \begin{exparts} \item The characteristic polynomial factors as $x^2-20x+75=(x-5)(x-15)$, so the eigenvalues are $\lambda_1=5$ and~$\lambda_2=15$. These are the associated eigenspaces. \begin{equation*} V_5=\set{k\colvec{1 \\ 2}\suchthat k\in\C} \quad V_{15}=\set{k\colvec{-2 \\ 1}\suchthat k\in\C} \end{equation*} For each eigenvalue, both the algebraic and geometric multiplicities are~$1$. \item The characteristic polynomial $x^3 - 6x^2 + 32$ factors into $(x+2)(x-4)^2$. The eigenvectors are $\lambda_1=-2$ and $\lambda_2=4$. Here are the associated eigenspaces. \begin{equation*} V_{-2}=\set{k\colvec{1 \\ 1 \\ 2}\suchthat k\in\C} \quad V_4=\set{k_1\colvec{1 \\ 1 \\ 0} +k_2\colvec{1 \\ 0 \\ -1} \suchthat k_1,k_2\in\C} \end{equation*} For $\lambda_1=-2$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=4$ the algebraic and geometric multiplicities are both~$2$. \item The characteristic polynomial $x^3 - 5x^2 + 8x - 4$ factors into $(x-1)(x-2)^2$. The eigenvalues are $\lambda_1=1$ and $\lambda_2=2$. Here are the associated eigenspaces. \begin{equation*} V_1=\set{k\colvec{-6 \\ 3 \\ 1}\suchthat k\in \C} \quad V_2=\set{k\colvec{1 \\ 0 \\ 0}\suchthat k\in\C} \end{equation*} For $\lambda_1=1$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=2$ the algebraic multiplicity is~$2$ but the geometric multiplicity is~$1$. \end{exparts} \end{ans} \end{ans} \begin{ans}{Five.II.3.28} \begin{ans}{Five.II.3.28} With respect to the natural basis $B=\sequence{1,x,x^2}$ With respect to the natural basis $B=\sequence{1,x,x^2}$
 ... @@ -2858,14 +2858,13 @@ It is a nontrivial subspace. ... @@ -2858,14 +2858,13 @@ It is a nontrivial subspace. \begin{proof} \begin{proof} %<*pf:EigSpaceIsSubSp> %<*pf:EigSpaceIsSubSp> To show that an eigenspace is a subspace, Notice first that notice first that $V_{\lambda}$ is not empty; it contains the zero $V_{\lambda}$ contains the zero vector since $t(\zero)=\zero$, which equals vector since $t(\zero)=\zero$, which equals $\lambda\cdot \zero$. $\lambda\cdot \zero$. So the eigenspace is a nonempty subset of the space. To show that an eigenspace is a subspace, What remains is to check closure of this set under linear combinations. what remains is to check closure of this set under linear combinations. Take $$\vec{\zeta}_1,\ldots,\vec{\zeta}_n\in V_{\lambda}$$ and then verify Take $$\vec{\zeta}_1,\ldots,\vec{\zeta}_n\in V_{\lambda}$$ and then \begin{align*} \begin{align*} t(\lincombo{c}{\vec{\zeta}}) t(\lincombo{c}{\vec{\zeta}}) &=c_1t(\vec{\zeta}_1)+\dots+c_nt(\vec{\zeta}_n) \\ &=c_1t(\vec{\zeta}_1)+\dots+c_nt(\vec{\zeta}_n) \\ ... @@ -3290,71 +3289,6 @@ In the next section we study matrices that cannot be diagonalized. ... @@ -3290,71 +3289,6 @@ In the next section we study matrices that cannot be diagonalized. \end{equation*} \end{equation*} \end{exparts} \end{exparts} \end{answer} \end{answer} \item % TODO: credit http://algebra.math.ust.hk/eigen/05_multiplicity/lecture2.shtml For each matrix, find the characteristic equation and the eigenvalues, and associated eigenspaces. Also find the algebraic and geometric multiplicities. \begin{exparts*} \partsitem $\begin{mat} 13 &-4 \\ -4 &7 \end{mat}$ \partsitem $\begin{mat} 1 &3 &-3 \\ -3 &7 &-3 \\ -6 &6 &-2 \end{mat}$ \partsitem $\begin{mat} 2 &3 &-3 \\ 0 &2 &-3 \\ 0 &0 &1 \end{mat}$ \end{exparts*} \begin{answer} \begin{exparts} \item The characteristic polynomial factors as $x^2-20x+75=(x-5)(x-15)$, so the eigenvalues are $\lambda_1=5$ and~$\lambda_2=15$. These are the associated eigenspaces. \begin{equation*} V_5=\set{k\colvec{1 \\ 2}\suchthat k\in\C} \quad V_{15}=\set{k\colvec{-2 \\ 1}\suchthat k\in\C} \end{equation*} For each eigenvalue, both the algebraic and geometric multiplicities are~$1$. \item The characteristic polynomial $x^3 - 6x^2 + 32$ factors into $(x+2)(x-4)^2$. The eigenvectors are $\lambda_1=-2$ and $\lambda_2=4$. Here are the associated eigenspaces. \begin{equation*} V_{-2}=\set{k\colvec{1 \\ 1 \\ 2}\suchthat k\in\C} \quad V_4=\set{k_1\colvec{1 \\ 1 \\ 0} +k_2\colvec{1 \\ 0 \\ -1} \suchthat k_1,k_2\in\C} \end{equation*} For $\lambda_1=-2$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=4$ the algebraic and geometric multiplicities are both~$2$. \item The characteristic polynomial $x^3 - 5x^2 + 8x - 4$ factors into $(x-1)(x-2)^2$. The eigenvalues are $\lambda_1=1$ and $\lambda_2=2$. Here are the associated eigenspaces. \begin{equation*} V_1=\set{k\colvec{-6 \\ 3 \\ 1}\suchthat k\in \C} \quad V_2=\set{k\colvec{1 \\ 0 \\ 0}\suchthat k\in\C} \end{equation*} For $\lambda_1=1$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=2$ the algebraic multiplicity is~$2$ but the geometric multiplicity is~$1$. \end{exparts} \end{answer} \item \item Find the characteristic equation, and the Find the characteristic equation, and the eigenvalues and associated eigenvectors for this matrix. eigenvalues and associated eigenvectors for this matrix. ... @@ -3697,11 +3631,76 @@ In the next section we study matrices that cannot be diagonalized. ... @@ -3697,11 +3631,76 @@ In the next section we study matrices that cannot be diagonalized. \end{equation*} \end{equation*} \end{exparts} \end{exparts} \end{answer} \end{answer} \item For each matrix, find the characteristic polynomial, and the eigenvalues and associated eigenspaces. Also find the algebraic and geometric multiplicities. \begin{exparts*} \partsitem $\begin{mat} 13 &-4 \\ -4 &7 \end{mat}$ \partsitem $\begin{mat} 1 &3 &-3 \\ -3 &7 &-3 \\ -6 &6 &-2 \end{mat}$ \partsitem $\begin{mat} 2 &3 &-3 \\ 0 &2 &-3 \\ 0 &0 &1 \end{mat}$ \end{exparts*} \begin{answer} \begin{exparts} \item The characteristic polynomial factors as $x^2-20x+75=(x-5)(x-15)$, so the eigenvalues are $\lambda_1=5$ and~$\lambda_2=15$. These are the associated eigenspaces. \begin{equation*} V_5=\set{k\colvec{1 \\ 2}\suchthat k\in\C} \quad V_{15}=\set{k\colvec{-2 \\ 1}\suchthat k\in\C} \end{equation*} For each eigenvalue, both the algebraic and geometric multiplicities are~$1$. \item The characteristic polynomial $x^3 - 6x^2 + 32$ factors into $(x+2)(x-4)^2$. The eigenvectors are $\lambda_1=-2$ and $\lambda_2=4$. Here are the associated eigenspaces. \begin{equation*} V_{-2}=\set{k\colvec{1 \\ 1 \\ 2}\suchthat k\in\C} \quad V_4=\set{k_1\colvec{1 \\ 1 \\ 0} +k_2\colvec{1 \\ 0 \\ -1} \suchthat k_1,k_2\in\C} \end{equation*} For $\lambda_1=-2$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=4$ the algebraic and geometric multiplicities are both~$2$. \item The characteristic polynomial $x^3 - 5x^2 + 8x - 4$ factors into $(x-1)(x-2)^2$. The eigenvalues are $\lambda_1=1$ and $\lambda_2=2$. Here are the associated eigenspaces. \begin{equation*} V_1=\set{k\colvec{-6 \\ 3 \\ 1}\suchthat k\in \C} \quad V_2=\set{k\colvec{1 \\ 0 \\ 0}\suchthat k\in\C} \end{equation*} For $\lambda_1=1$ the algebraic and geometric multiplicities are both~$1$. For $\lambda_2=2$ the algebraic multiplicity is~$2$ but the geometric multiplicity is~$1$. \end{exparts} \end{answer} \recommended \item \recommended \item Let $$\map{t}{\polyspace_2}{\polyspace_2}$$ be Let $$\map{t}{\polyspace_2}{\polyspace_2}$$ be this linear map. \begin{equation*} \begin{equation*} a_0+a_1x+a_2x^2\mapsto a_0+a_1x+a_2x^2\mapsto (5a_0+6a_1+2a_2)-(a_1+8a_2)x+(a_0-2a_2)x^2. (5a_0+6a_1+2a_2)-(a_1+8a_2)x+(a_0-2a_2)x^2 \end{equation*} \end{equation*} Find its eigenvalues and the associated eigenvectors. Find its eigenvalues and the associated eigenvectors. \begin{answer} \begin{answer} ... ...