# Questions tagged [eigenvalues-eigenvectors]

Eigenvalues are numbers associated to a linear operator from a vector space $V$ to itself: $\lambda$ is an eigenvalue of $T\colon V\to V$ if the map $x\mapsto \lambda x-Tx$ is not injective. An eigenvector corresponding to $\lambda$ is a non-trivial solution to $\lambda x - Tx = 0$.

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### range of $\lambda I - T$ and the null space of $\bar \lambda I - T^*$

I have a few questions about the following exercise in Stein's Real analysis. Exercise 29 Let $T$ be a compact operator on a Hilbert Space $\mathcal H$ and assume $\lambda \neq 0$. (a) Show that the ...
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### Is it possible to show that the eigenbasis is the most 'optimal' basis for a given linear transformation (if it exists)? How would you measure this?

Given a linear transformation, a coordinate transformation to the eigenbasis (if it exists) seems the most optimal and easiest to calculate. An eigenvector expressed in other coordinates always seems ...
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### If every eigenvector of $\Phi + \Phi^{*}$ is also an eigenvector of $\Phi - \Phi^{*}$, $\Phi$ is normal.

Let $V$ be an inner product space with finite dimension and $\Phi: V \rightarrow V$ a homomorphism. If every eigenvector of $\Phi+\Phi^{*}$ is also an eigenvector of $\Phi-\Phi^{*}$, how can we ...
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### Linear operator $A: \mathbb R^n \to \mathbb R^n$

Linear operator $A: \mathbb R^n \to \mathbb R^n$ such as $A^3$ — projection operator. What eigenvalues could $A$ has? Is it correct that $A$ will have diagonal matrix in some $\mathbb R^n$ basis? My ...
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### Why does asymptotically stable fixed point become saddle structure in traveling wave coordinates?

Suppose the reaction-diffusion system Suppose the dynamical system \begin{align*} v_t &= f(v, w) \\ w_t &= g(v, w) \end{align*} where $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$, has an ...
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### Find a basis for the orthogonal to span (V1,V2) [closed]

v1=2 1 v2=0 1 0 Find a basis for the orthogonal complement to span (v1,V2) where:
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