# Tagged Questions

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### Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
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### Solve a System with Variable

Given these matrices, how does one find two real solutions? $dx/dt$ = $\begin{bmatrix} 3 & -5\\ 5 & 3 \end{bmatrix}x$ with $x(0) = \begin{bmatrix} 2\\ -3 \end{bmatrix}$
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### Same eigenvalues, different eigenvectors but orthogonal

I am using a two different computational libraries to calculate the eigenvectors and eigenvalues of a symmetric matrix. The results show that the eigenvalues calculated with both libraries are exactly ...
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### Common eigenvector of two linear transformation matrices

I have two linear transformation matrices \begin{pmatrix} 3 & 2 \\ -2 & 1 \end{pmatrix} and \begin{pmatrix} 1-a & -a \\ a & 1 \end{pmatrix} How to find out what the value of ...
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### generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors

I am trying to find a generalized eigenvector in this problem. (I understand the general theory goes much deeper, but we are only responsible for a limited number of cases.) I have found ...
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### Hermitian matrices [duplicate]

Suppose we have a hermitian matrix $H$, and a matrix $A$ composed of eigenvectors of $H$, such that $\langle A_i \mid A_i \rangle =1$, where $A_i$ is the $i$-th column of matrix H. How to prove ...
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### Calculate eigenvectors

I am given the $2\times2$ matrix $$A = \begin {bmatrix} -2&-1 \\\\ 15&6 \ \end{bmatrix}$$ I calculated the Eigenvalues to be 3 and 1. How do I find the vectors? If I plug the value back into ...
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### Hermitian Matrices are Diagonalizable

I am trying to prove that Hermitian Matrices are diagonalizable. I have already proven that Hermitian Matrices have real roots and any two eigenvectors associated with two distinct eigen values are ...
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### Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that Î» = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
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### A Has characteristic polynomial that can be reduced to linear products $\Rightarrow$ A similar to upper triangular Matrix

Prove that if $A\in M_{n}\left(\mathbb{F}\right)$ matrix with a characteristic polynomial that can be written as a product of linear elements (?) ...
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### Generalised eigenvalue is eigenvalue if it is in the field

I would like to prove the following assertion: Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
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### Bounding the smallest eigenvalue of an ergodic Markov Chain

I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
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### An eigenvector is a non-zero vector such that…

Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...