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Let $A \in M(n, \mathbb{F})$ have an eigenvalue $\lambda$ with corresponding eigenvector $v$. True or False
Let $A \in M(n, \mathbb{F})$ have an eigenvalue $\lambda$ with corresponding eigenvector $v$. True or FalseLet $A \in M(n, \mathbb{F})$ have an eigenvalue $\lambda$ with corresponding eigenvector $v$. True or False a) $-v$ is an eigenvector of $-A\ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:
Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions:Let $C$ be a $2 \times 2$ matrix of real numbers. Give a proof or counterexample to each of the following assertions: a) $\operatorname{det}\left ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the assertions that are correct.
Let $L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the assertions that are correct.Let $L$ be an $n \times n$ matrix with real entries and let $\lambda$ be an eigenvalue of $L$. In the following list, identify all the asserti ...
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A $3 \times 3$ real matrix need not have any real eigenvalues.
A $3 \times 3$ real matrix need not have any real eigenvalues.True or False - and Why? a) A $3 \times 3$ real matrix need not have any real eigenvalues. b) If an $n \times n$ matrix $A$ is invertible, then ...
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Let $A$ be an invertible matrix. If $\mathbf{V}$ is an eigenvector of $A$, show it is also an eigenvector of both $A^{2}$ and $A^{-2}$. What are the corresponding eigenvalues?Let $A$ be an invertible matrix. If $\mathbf{V}$ is an eigenvector of $A$, show it is also an eigenvector of both $A^{2}$ and $A^{-2}$. What ...
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Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and $A \vec{w}$ are orthogonal.Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and ...
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Let $A$ be a square matrix. Proof or Counterexample. If $A$ is diagonalizable, then so is $A^{2}$.
Let $A$ be a square matrix. Proof or Counterexample. If $A$ is diagonalizable, then so is $A^{2}$.Let $A$ be a square matrix. Proof or Counterexample. a) If $A$ is diagonalizable, then so is $A^{2}$. b) If $A^{2}$ is diagonalizable, then so ...
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Let $a, b, c, d$, and $e$ be real numbers. For each of the following matrices, find their eigenvalues, corresponding eigenvectors, and orthogonal matrices that diagonalize them.Let $a, b, c, d$, and $e$ be real numbers. For each of the following matrices, find their eigenvalues, corresponding eigenvectors, and orthogonal ...
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Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$
Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 2\end{array}\right)$Find an orthogonal matrix $R$ that diagonalizes $A:=\left(\begin{array}{rrr}1 &amp; -1 &amp; 0 \\ -1 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 2\end{array ... close 0 answers 2 views An \(n \times n$ matrix is called nilpotent if $A^{k}$ equals the zero matrix for some positive integer $k$. (For instance, $\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)$ is nilpotent.)An $n \times n$ matrix is called nilpotent if $A^{k}$ equals the zero matrix for some positive integer $k$. (For instance, $\left(\begin{array} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)$ - Diagonalize $A$.
Let $A=\left(\begin{array}{cc}a & b-a \\ 0 & b\end{array}\right)$ - Diagonalize $A$.Let $A=\left(\begin{array}{cc}a &amp; b-a \\ 0 &amp; b\end{array}\right)$ a) Diagonalize $A$. b) Use this to compute $A^{k}$ for any integer $... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If a matrix \(A$ is diagonalizable, show that for any scalar $c$ so is the matrix $A+c I$.
If a matrix $A$ is diagonalizable, show that for any scalar $c$ so is the matrix $A+c I$.If a matrix $A$ is diagonalizable, show that for any scalar $c$ so is the matrix $A+c I$. ...
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Diagonalize the matrix $A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right)$
Diagonalize the matrix $A=\left(\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right)$Diagonalize the matrix \ A=\left(\begin{array}{lll} 1 &amp; 0 &amp; 2 \\ 0 &amp; 1 &amp; 0 \\ 2 &amp; 0 &amp; 1 \end{array}\right) \ by finding the ...
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Let $A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)$.
Let $A=\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 1 & 2 \\ 1 & 1 & 2\end{array}\right)$.Let $A=\left(\begin{array}{lll}1 &amp; 1 &amp; 2 \\ 1 &amp; 1 &amp; 2 \\ 1 &amp; 1 &amp; 2\end{array}\right)$. a) What is the dimension of the image ...
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Let $A$ be an $n \times n$ real self-adjoint matrix and $\mathbf{v}$ an eigenvector with eigenvalue $\lambda$. Let $W=\operatorname{span}\{\mathbf{v}\}$.
Let $A$ be an $n \times n$ real self-adjoint matrix and $\mathbf{v}$ an eigenvector with eigenvalue $\lambda$. Let $W=\operatorname{span}\{\mathbf{v}\}$.Let $A$ be an $n \times n$ real self-adjoint matrix and $\mathbf{v}$ an eigenvector with eigenvalue $\lambda$. Let $W=\operatorname{span}\{\m ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If \(A$ and $B$ can be simultaneously diagonalized, show that $A B=B A$.
If $A$ and $B$ can be simultaneously diagonalized, show that $A B=B A$.Two matrices $A, B$ can be simultaneously diagonalized if there is an invertible matrix that diagonalizes both of them. In other words, if there is ...
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What can you say about the eigenvalues and eigenvectors of $A^{-1} ?$ Justify your response.
What can you say about the eigenvalues and eigenvectors of $A^{-1} ?$ Justify your response.Let $A$ be an invertible matrix with eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{k}$ and corresponding eigenvectors $\vec{v}_{1}, \ve ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A$ be a square matrix. If the eigenvectors $v_{1}, \ldots v_{k}$ have distinct eigenvalues, show that these vectors are linearly independent.
Let $A$ be a square matrix. If the eigenvectors $v_{1}, \ldots v_{k}$ have distinct eigenvalues, show that these vectors are linearly independent.Let $A$ be a square matrix. If the eigenvectors $v_{1}, \ldots v_{k}$ have distinct eigenvalues, show that these vectors are linearly independent. ...
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Give an example of a matrix $A$ with the following three properties:
Give an example of a matrix $A$ with the following three properties:Give an example of a matrix $A$ with the following three properties: i). $A$ has eigenvalues $-1$ and 2 . ii). The eigenvalue $-1$ has eigenve ...
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Compute the determinant of $A^{10}+A$.
Compute the determinant of $A^{10}+A$.a) Find a $2 \times 2$ real matrix $A$ that has an eigenvalue $\lambda_{1}=1$ with eigenvector $E_{1}=$ $\left(\begin{array}{l}1 \\ 2\end{arr ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 [Frobenius] Let \(A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined.
[Frobenius] Let $A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined.Frobenius Let $A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined. Use the obvious \ \operatorname{dim}\left(\left ...
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Let $U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show that
Let $U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show thatLet $U \subset V$ and $W$ be finite dimensional linear spaces and $L: V \rightarrow W$ a linear map. Show that \ \operatorname{dim}\left(\left. ...
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Let $A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}$ and $B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}$.
Let $A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}$ and $B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}$.Let $A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}$ and $B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}$. Prove that $\operatorname{rank} A+\o ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A(t)$ be a family of real square matrices depending smoothly on the parameter $t$.
Let $A(t)$ be a family of real square matrices depending smoothly on the parameter $t$.Let $A(t)$ be a family of real square matrices depending smoothly on the parameter $t$. a) Find a formula for the derivative of $A^{2}(t)$. b) ...
Let $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible.
Let $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible.Let $A(t)$ be a family of invertible real matrices depending on the real parameter $t$ and assume they are invertible. Show that the inverse matri ...