# Recent questions tagged eigenvalue

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When is a matrix $M$ normal?
When is a matrix $M$ normal?When is a matrix $M$ normal? ...
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A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal.
A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal.A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal. ...
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Let $A$ be a square matrix of real numbers whose columns are (non-zero) orthogonal vectors.
Let $A$ be a square matrix of real numbers whose columns are (non-zero) orthogonal vectors.Let $A$ be a square matrix of real numbers whose columns are (non-zero) orthogonal vectors. a) Show that $A^{T} A$ is a diagonal matrix - whose in ...
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Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix with the property that its absolute row sums are at most 1 , that is, $\left|a_{i 1}\right|+\cdots+\left|a_{\text {in }}\right| \leq 1$ for all $i=1, \ldots, n$.Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix with the property that its absolute row sums are at most 1 , that is, $\left|a_{i 1}\right ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 [SPECTRAL MAPPING THEOREM] Let \(A$ be a square matrix.
[SPECTRAL MAPPING THEOREM] Let $A$ be a square matrix.SPECTRAL MAPPING THEOREM Let $A$ be a square matrix. a) If $A(A-I)(A-2 I)=0$, show that the only possible eigenvalues of $A$ are $\lambda=0$ ...
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Identify all possible eigenvalues of an $n \times n$ matrix $A$ that satisfies the matrix equation: $A-2 I=-A^{2}$. Justify your answer.
Identify all possible eigenvalues of an $n \times n$ matrix $A$ that satisfies the matrix equation: $A-2 I=-A^{2}$. Justify your answer.a) Identify all possible eigenvalues of an $n \times n$ matrix $A$ that satisfies the matrix equation: $A-2 I=-A^{2}$. Justify your answer. b) M ...
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Let $A$ be a square matrix and let $\|B\|$ be any norm on matrices [one example is $\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right]$.
Let $A$ be a square matrix and let $\|B\|$ be any norm on matrices [one example is $\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right]$.Let $A$ be a square matrix and let $\|B\|$ be any norm on matrices one example is $\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right$. To wha ...
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Let $\alpha_{1}, \ldots, \alpha_{n}$ be positive real numbers and let $A=\left(a_{i j}\right)$ where $a_{i j}=\alpha_{i} / \alpha_{j} .$ Say as much as you can about the eigenvalues and eigenvectors of $A$.Let $\alpha_{1}, \ldots, \alpha_{n}$ be positive real numbers and let $A=\left(a_{i j}\right)$ where $a_{i j}=\alpha_{i} / \alpha_{j} .$ Say as ...
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Show that $p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{0}$ is the characteristic polynomial of the matrix
Show that $p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{0}$ is the characteristic polynomial of the matrixShow that $p(\lambda)=\lambda^{n}+a_{n-1} \lambda^{n-1}+\cdots+a_{0}$ is the characteristic polynomial of the matrix \ A=\left(\begin{array}{ccccc} ...
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Let $Z$ be a complex square matrix whose self-adjoint part is positive definite, so $Z+Z^{*}$ is positive definite.
Let $Z$ be a complex square matrix whose self-adjoint part is positive definite, so $Z+Z^{*}$ is positive definite.Let $Z$ be a complex square matrix whose self-adjoint part is positive definite, so $Z+Z^{}$ is positive definite. a) Show that the eigenvalues o ...
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Let $A$ and $B$ be $n \times n$ complex matrices that commute: $A B=B A$. If $\lambda$ is an eigenvalue of $A$, let $\mathcal{V}_{\lambda}$ be the subspace of all eigenvectors having this eigenvalue.Let $A$ and $B$ be $n \times n$ complex matrices that commute: $A B=B A$. If $\lambda$ is an eigenvalue of $A$, let $\mathcal{V}_{\lambda ... close 0 answers 72 views Let \(M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant $c$. Which (if any) any of the vectorsLet $M$ be a $2 \times 2$ matrix with the property that the sum of each of the rows and also the sum of each of the columns is the same constant \ ...
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Let $A$ be a square matrix and $p(\lambda)$ any polynomial. If $\lambda$ is an eigenvalue of $A$, show that $p(\lambda)$ is an eigenvalue of the matrix $p(A)$ with the same eigenvector.Let $A$ be a square matrix and $p(\lambda)$ any polynomial. If $\lambda$ is an eigenvalue of $A$, show that $p(\lambda)$ is an eigenvalue of ...
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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 \(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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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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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 Every real upper triangular \(n \times n$ matrix $\left(a_{i j}\right)$ with $a_{i i}=1, i=1,2, \ldots, n$ is invertible. Proof or counterexample.
Every real upper triangular $n \times n$ matrix $\left(a_{i j}\right)$ with $a_{i i}=1, i=1,2, \ldots, n$ is invertible. Proof or counterexample.Every real upper triangular $n \times n$ matrix $\left(a_{i j}\right)$ with $a_{i i}=1, i=1,2, \ldots, n$ is invertible. Proof or counterexample ...
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Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero real vector.
Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero real vector.Let $A$ be the rank one $n \times n$ matrix $A=\left(v_{i} v_{j}\right)$, where $\vec{v}:=\left(v_{1}, \ldots, v_{n}\right)$ is a non-zero rea ...
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Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the image of $A$.
Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the image of $A$.Let $A=\left(a_{i j}\right)$ be an $n \times n$ matrix whose rank is 1 . Let $v:=\left(v_{1}, \ldots, v_{n}\right) \neq 0$ be a basis for the im ...
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Let $A$ be a square real matrix all of whose eigenvalues are zero. Show that $A$ is diagonalizable (that is, similar to a possibly comples diagonal matrix) if and only if $A=0$.Let $A$ be a square real matrix all of whose eigenvalues are zero. Show that $A$ is diagonalizable (that is, similar to a possibly comples diagona ...
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Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero.
Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero.Give an example of a square real matrix that has rank 2 and all of whose eigenvalues are zero. ...
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Let $A$ and $B$ be $n \times n$ matrices with the property that $A B=0 .$ For each of the following give a proof or counterexample.
Let $A$ and $B$ be $n \times n$ matrices with the property that $A B=0 .$ For each of the following give a proof or counterexample.Let $A$ and $B$ be $n \times n$ matrices with the property that $A B=0 .$ For each of the following give a proof or counterexample. a) Every ...
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Let $A$ be a square real (or complex) matrix. Then $A$ is invertible if and only if zero is not an eigenvalue. Proof or counterexample.
Let $A$ be a square real (or complex) matrix. Then $A$ is invertible if and only if zero is not an eigenvalue. Proof or counterexample.Let $A$ be a square real (or complex) matrix. Then $A$ is invertible if and only if zero is not an eigenvalue. Proof or counterexample. ...
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Define what a symmetric matrix is
Define what a symmetric matrix isDefine what a symmetric matrix is ...
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If the eigenvalues of $\mathbf{A}$ are $100,-50,3 \pm 5 \sqrt{-1}$, then find a lower bound on cond $(\mathbf{A})$
If the eigenvalues of $\mathbf{A}$ are $100,-50,3 \pm 5 \sqrt{-1}$, then find a lower bound on cond $(\mathbf{A})$ Given a nonsingular $n \times n$ matrix $\mathbf{A}$, and $n \times 1$ vectors $\mathbf{x}, \hat{\mathbf{x}}, \mathbf{b}=\mathbf{A x}$, and $\hat{\mat ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find a better bound for $$\frac{\|\mathbf{x}-\hat{\mathbf{x}}\|}{\|\mathbf{x}\|} \times \frac{\|\mathbf{b}\|}{\|\mathbf{b}-\hat{\mathbf{b}}\|}$$ 1 answer 226 views Find a better bound for $$\frac{\|\mathbf{x}-\hat{\mathbf{x}}\|}{\|\mathbf{x}\|} \times \frac{\|\mathbf{b}\|}{\|\mathbf{b}-\hat{\mathbf{b}}\|}$$Given a nonsingular$n \times n$matrix$\mathbf{A}$, and$n \times 1$vectors$\mathbf{x}, \hat{\mathbf{x}}, \mathbf{b}=\mathbf{A x}$, and$\hat{\mat ...
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Prove that if A is positive semi-definite and $\epsilon>0$, then $\mathbf{A}+\epsilon \mathbf{I}$ is positive definite.
Prove that if A is positive semi-definite and $\epsilon>0$, then $\mathbf{A}+\epsilon \mathbf{I}$ is positive definite.Prove that if A is positive semi-definite and $\epsilon&gt;0$, then $\mathbf{A}+\epsilon \mathbf{I}$ is positive definite. ...
Prove that for  $\mathbf{A} \in \mathbb{R}^{m \times n}$, $\mathbf{A}^{\top} \mathbf{A}$ is positive semi-definite. If $\operatorname{null}(\mathbf{A})=\{0\}$, then $\mathbf{A}^{\top} \mathbf{A}$ is positive definite.Prove that for &nbsp;$\mathbf{A} \in \mathbb{R}^{m \times n}$, $\mathbf{A}^{\top} \mathbf{A}$ is positive semi-definite. If $\operatorname{null}(\math ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 A symmetric matrix A is positive semi-definite if 1 answer 208 views A symmetric matrix A is positive semi-definite ifA symmetric matrix A is positive semi-definite if ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 A matrix$\mathbf{A} \in \mathbb{R}^{n \times n}$is said to be symmetric if 1 answer 276 views A matrix$\mathbf{A} \in \mathbb{R}^{n \times n}$is said to be symmetric ifA matrix$\mathbf{A} \in \mathbb{R}^{n \times n}$is said to be symmetric if ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 A matrix$\mathbf{Q} \in \mathbb{R}^{n \times n}$is said to be orthogonal if 1 answer 200 views A matrix$\mathbf{Q} \in \mathbb{R}^{n \times n}$is said to be orthogonal ifA matrix$\mathbf{Q} \in \mathbb{R}^{n \times n}$is said to be orthogonal if ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$\mathbf{x}$be an eigenvector of$\mathbf{A}$with corresponding eigenvalue$\lambda$. Then 1 answer 211 views Let$\mathbf{x}$be an eigenvector of$\mathbf{A}$with corresponding eigenvalue$\lambda$. ThenLet$\mathbf{x}$be an eigenvector of$\mathbf{A}$with corresponding eigenvalue$\lambda$. Then ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 We say that a nonzero vector$\mathrm{x} \in \mathbb{R}^{n}$is an eigenvector of A corresponding to eigenvalue$\lambda$if 1 answer 235 views We say that a nonzero vector$\mathrm{x} \in \mathbb{R}^{n}$is an eigenvector of A corresponding to eigenvalue$\lambda$ifFor a square matrix$\mathbf{A} \in \mathbb{R}^{n \times n}$, there may be vectors which, when$\mathbf{A}\$ is applied to them, are simply scaled by s ...