# Tagged Questions

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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### What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
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### Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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### What do the eigenvalues/vectors of a metric describe?

Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$. What does the eigenspectrum of this matrix say about the metric $d$? ...
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### Coefficients of spherical solution to Laplace's equation with difficult Robin boundary conditions

I'm trying to solve Laplace's equation in an (axisymmetric) external spherical domain. The controlling equation is: $$\nabla^2 f = 0$$ $f$ must dissappear at infinity, and at the surface of the ...
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### finding the algebraic dimension of $\ell^p$ spaces

I want to know "how we can find the algebraic dimension(the cardinal number of the Hamel basis) for $\ell^p$ spaces." What can we say about $\ell^p(I)$, where $I$ is an infinite set?\ Moreover, for ...
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### Reference for Generalized Eigenvectors

I am looking for references on generalized eigenvectors and Jordan matrix representation. I would like a brief but complete introduction of this concepts with a nice treatment of the most important ...
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### “Rank-K Correction” of a matrix and significance?

Today my studies led me to read about the matrix inversion lemma, which Wikipedia introduces as follows: In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max ...
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### A problem on 0-1 matrices.

Given a 0-1 matrix $A$, is there an efficient way to find all 0-1 vectors $x$ such that $Ax = v$ where the entries of $v$ belong to a set $\{a,b\} \subseteq \mathbb{Z}$ of size $2$? Note that $v$ is ...
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### Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$F:S\times T\longrightarrow K$$ where $K$ is a topological field ...
I read about rank nullity theorem (with proof) but then tried to prove it in different way. Please can you read my proof and tell me if it is correct? The rank nullity theorem: If $T:V\to W$ is a ...
I am looking for a (simple) upper bound for the smallest eigenvalue of an $n\times n$ matrix, involving determinant or trace or something else that can be easily computed. I've got an upper bound from ...