All Questions
Tagged with generalizedeigenvector or generalized-eigenvector
233 questions
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Eigenvector of GSE matrix with right properties
I encountered a weird problem when trying to study a few properties of eigenvectors of matrices sampled from the Gaussian symplectic ensemble (GSE). I have encountered this while trying to understand ...
- 310
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71
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Ratio of eigenvector amplitudes
I am studying a generalized eigenvalue problem which can be partitioned as
$$
\left[ {\begin{array}{*{20}{c}}
{{A_{11}}}&{{A_{12}}}\\
{{A_{21}}}&{{A_{22}}}
\end{array}} \right]\left[ {\begin{...
- 1
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12
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Discrepancy in inverse calculated using GHEP and HEP
Say we have a matrix $A = L + \beta^{2} M$, where $\beta$ is a real scalar. The matrices $L$ and $M$ are symmetric positive semi-definite and symmetric positive definite respectively. I am interested ...
- 31
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53
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Can Krylov subspace methods like Lanczos be used for generalized eigenvalue problems with defective eigenvalues?
I’m studying generalized eigenvalue problems of the form $A x = \lambda B x$, and I’m particularly interested in defective eigenvalues—those with fewer linearly independent eigenvectors than their ...
- 43
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55
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Is there a name for this kind of generalized eigenvalue problem
I've been thinking about a particular kind of "generalized eigenvalue", in the following sense:
Suppose we have two vector spaces $V_1$ and $V_2$, and let $A$ be a matrix over $V_1$. The &...
- 121
3
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1
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125
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Basis for generalized eigenspaces with tensor products and a suitable coproduct.
This question comes from when I was learning about quantum groups. I want to know the basis of generalized eigenspaces with tensor products
Assume that $\phi_m(m\ge 0)$ are operators with $\phi_n\...
- 1,238
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votes
1
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55
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multiplicity of an eigenvalue equals number of times on diagonal
I am trying to understand the proof of Theorem 8.31 in Linear Algebra Done Right, 4th edition. Below is the theorem.
Suppose 𝐅 = 𝐂 and $𝑇 ∈ L(𝑉)$. Suppose $𝑣_1,...,𝑣_𝑛$ is a basis of $𝑉$ ...
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Generalized eigenvectors of operators
I'm reading Brian Hall's Quantum theory for Mathematicians, and when discussing the position operator $f(t)\to t f(t)$, for each $f\in L^2([0,1])$, he notes that there are no true eigenvectors, but ...
- 3,355
1
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1
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74
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Issue with finding the Jordan normal form and the Jordan basis
I am trying to figure out what is going wrong when computing the Jordan normal form for $$A=\begin{bmatrix} 3 & 0 & 1 & -1 \\ 0 & 3 & 1 & 0\\ 0 & 0 & 3 & 1\\ 0 &...
- 13
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79
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How to choose the right generalized eigenvector for Jordan normal form
Given $A = \begin{bmatrix}
0 & -4 & 0 & -1\\
0 & 2 & 0 & 0\\
0 & 0 & 0 & 0\\
4 & 8 & -12 & 4\end{bmatrix}$
With a characteristic ...
- 29
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1
answer
163
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Proof that a generalized eigenspace is disjoint from the span of all other generalized eigenspaces
I am trying to understand the proof of Theorem 2 from Paul R. Halmos's Finite-Dimensional Vector Spaces ($\S{58}$ Jordan form, pp 113-114) which is part of the proof of generalized eigenspace ...
- 51
2
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1
answer
122
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A linear map preserves generalized eigenspaces.
Let $V, W$ be vector spaces over a field $K$ of finite dimension. $f: V\to V$ and $g: W\to W$ are linear transformations. $\phi: V\to W$ is a surjective linear map such that $\phi f=g\phi$．$V\supseteq ...
- 53
2
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1
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84
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Fisher's Linear Discriminant analysis scatter matrix
I am reading about Linear Discriminant Analysis, especially Fisher's discriminat function which solves and optimization problem $Max\dfrac{w^TS_Bw}{w^TS_Ww}$. Lets say i have a two class problem. The ...
- 2,712
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96
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Equivalent matrix given a constraint on elements
Assume there is a matrix which has elements with a normal distribution. How to find a matrix which every element can only be -1, 0, or 1 such that it has closest eigenvalues to the original matrix. ...
- 303
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1
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99
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Eigenvalue sensitivities w.r.t. matrix entries
Consider the matrix valued function $A: \mathbb R \rightarrow \mathbb R^{n \times n}$. For a particular $x \in \mathbb R$, $A(x)$ has Eigenvalues $\lambda_1, \dots, \lambda_n$. Is it possible to ...
- 514