# Tagged Questions

For specific question about eigenvectors of a matrix or a linear operator.

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### what would be the eigen vector for this value?

So I have a $3\times 3$ matrix $$A=\begin{pmatrix} 2&1&1\\ 1&2&1\\ 1&1&2 \end{pmatrix}.$$ My instructions are to find the eigenvalues and eigenvectors of the matrix. For each ...
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### Updating the first k eigenpairs of L with a low rank perturbation

$L$ is a M-matrix, it's symmetric, semi-definite and the spectral radius is less than 1. Apparently, this is a normalized Laplacian Matrix of a graph. Given the first k eigenpair of ...
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### Eigenvectors are linearly independent?

Theorem: Eigenvectors corresponding to distinct eigenvalues are linearly independent. Could someone give me a geometric interpretation of the theorem? Thanks!
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### Meaning of eigenvalues/eigenvectors of matrices

I am supposed to find the geometric meaning of eigenvalues/eigenvectors of certain matrices such as reflections about x= y, rotaions, shears, etc. How would I go about this? The first question is a ...
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### Matrix Decomposition with restriction on eigenvalues

Let $B$ be an invertible $n\times n$ complex matrix. Prove that there exist $n\times n$ complex matrices $A, C$ such that the following three conditions are satisfied simultaneously: (i) $B = AC$ ...
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### Is the orthogonality of eigenvectors preserved in a rank deficient Hermitian matrix?

If $\boldsymbol{A}$ is a rank deficient Hermitian matrix are the following true? 1) Is $<\boldsymbol{x}_j,\boldsymbol{x_k}>=0$ when $\lambda_j=\lambda_k$? 2) Is ...
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### Eigenspace definition

This is the definition I have for eigenspace: Let $\lambda$ be an eigenvalue of $A \in Mn(\mathbb C)$. Then {$v \in \mathbb C^n|Av= \lambda v$}$=W_\lambda$ is a subspace of $\mathbb C^n$, called ...
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### Intuition behind independence of eigenvectors?

Theorem 6.14: Eigenvectors corresponding to distinct eigenvalues of A are linearly independent. My prof already gave us a proof of the theorem, so I'm not looking for another one. Could someone ...
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### Proof that Markov Chains converges to the stationary distribution

Let $P$ is a transition matrix of a Markov Chain, which is irreducible, aperiodic and lets assume $\pi$ is its stationary distribution: $\pi = \pi P$. Does anyone knows the proof for the following ...
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### Eigenspaces: Intuition behind geometric multiplicity $\leq$ algebraic multiplicity?

Theorem 6.6: Let $A$ be a square matrix, let $\gamma$ be an eigenvalue of $A$ with multiplicity $m$. Then the dimension of the eigenspace of $A$ corresponding to the eigenvalue $\gamma$ is less ...
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### Eigenvectors and values of nearly identical symmetrical matrices

I am given 2 matrices which have the following traits: Let $A$ and $B$ be those matrices and $a_{i,j}$ and $b_{i,j}$ be the entries of both matrices. There are 2 disjoint subsets of the indexes, let ...
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### Practical applications of eigenanalysis

I have always wondered: what are practical applications of finding (and using) eigenvalues and eigenvectors of matrices? I'm asking because i studied this at school, and at the time these things were ...
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### If $D$ is the operator of differentiation, prove $D^{2}$ is a self adjoint linear operator on V and find all its eigenvalues and eigenvectors

Suppose $V$ is the space of infinitely differentiable complex valued functions $f$ on $[0,\pi]$ such that $D^{2k+1}f(0) = 0 = D^{2k+1}f(\pi)$ for all integers $k \geq 0$. Then V is a complex IPS with ...
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### A question about the eigenvector and the basis

Let $(1, 0, 0)^T$ and $(0, -1, 1)^T$ be eigenvector of a 3x3 matrix $A$ with eigenvalue 1 and $(-2, -2, 1)^T$ be an eigenvector of $A$ with eigenvalue 2.Put $e_3=(0, 0, 1)^T$. Find eigenvector $v$ ...
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### Question about $(A - \lambda I_A)\vec{x} = 0$.

Finding a solution to $C\vec{x} = (A - \lambda I_A)\vec{x} = 0$ is the equivalent of considering the determinant of $C$ when it is zero. This means the matrix is linearly dependent and has infinite ...
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### Show that a transformation matrix is equal to the martix of eigenvectors

The real symmetric $3\times 3$ matrix $A$ has unit eigenvectors $\mathbf x_i$, $i=1,2,3$. Thus we have, $A\mathbf x_i=\lambda_i \mathbf x_i$. A $3\times 3$ matrix $C$ takes a vector in the ...