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Recent questions tagged eigenvalue


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When is a matrix \(M\) normal?
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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.

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Let \(A\) be a square matrix of real numbers whose columns are (non-zero) orthogonal vectors.
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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. a) Show that \(A^{T} A\) is a diagonal matrix - whose in ...
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[SPECTRAL MAPPING THEOREM] Let \(A\) be a square matrix.
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[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.

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Let \(A\) be a square matrix and let \(\|B\|\) be any norm on matrices [one example is \(\left.\|B\|=\max _{i, j}\left|b_{i j}\right|\right]\).
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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
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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 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.
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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. a) Show that the eigenvalues o ...
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Let \(A \in M(n, \mathbb{F})\) have an eigenvalue \(\lambda\) with corresponding eigenvector \(v\). True or False
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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\ ...
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A \(3 \times 3\) real matrix need not have any real eigenvalues.
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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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If a matrix \(A\) is diagonalizable, show that for any scalar \(c\) so is the matrix \(A+c I\).
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If \(A\) and \(B\) can be simultaneously diagonalized, show that \(A B=B A\).
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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.
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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 ...
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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:
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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: i). \(A\) has eigenvalues \(-1\) and 2 . ii). The eigenvalue \(-1\) has eigenve ...
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Compute the determinant of \(A^{10}+A\).
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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 ...
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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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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.
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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.
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Define what a symmetric matrix is
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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 $(\mathbf{A})$ 
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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 $(\mathbf{A})$ 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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Find a better bound for $$ \frac{\|\mathbf{x}-\hat{\mathbf{x}}\|}{\|\mathbf{x}\|} \times \frac{\|\mathbf{b}\|}{\|\mathbf{b}-\hat{\mathbf{b}}\|} $$

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Prove that if A is positive semi-definite and $\epsilon>0$, then $\mathbf{A}+\epsilon \mathbf{I}$ is positive definite.
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A symmetric matrix A is positive semi-definite if
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A symmetric matrix A is positive semi-definite ifA symmetric matrix A is positive semi-definite if ...
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A matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ is said to be symmetric if
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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 ...
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A matrix $\mathbf{Q} \in \mathbb{R}^{n \times n}$ is said to be orthogonal if
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Let $\mathbf{x}$ be an eigenvector of $\mathbf{A}$ with corresponding eigenvalue $\lambda$. Then
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We say that a nonzero vector $\mathrm{x} \in \mathbb{R}^{n}$ is an eigenvector of A corresponding to eigenvalue $\lambda$ if
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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 ...
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What’s an eigenvalue? What about an eigenvector?
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What’s an eigenvalue? What about an eigenvector? What’s an eigenvalue? What about an eigenvector? ...
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asked in Data Science & Statistics Sep 5, 2020
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