# Tagged Questions

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I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the ...
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### Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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### Probablistic bound for $\|RR^TM\|$ for uniformly random orthonormal matrix $R$

I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because ...
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### Should I change my Linear Algebra Textbook?

I know there are many questions related to linear algebra, but the textbook I'm using is not that widely used as other books, I guess. My university uses 'Finite-Dimensional Linear Algebra' by Mark ...
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### When the sum of coefficients of two linear combinations are equal.

I recently was looking a set of polynomials (the Legendre polynomials up to degree $n$) that form a basis for the space of polynomials $\{a_{0} + a_{1}x + \dots + a_{n}x^{n}: a_{i} \in \mathbb{R}\}$ ...
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### What is a good reference for learning about induced norms?

Wikipedia tells me a little about it. Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces". Which makes me want to ask: what's a good ...
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### complexity of solving $n \times n$ rank deficient linear system

I think it is known that given a nonsingular $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$, solving a linear system $Ax =b$ for $x$ can be done in $O(n^3)$ steps. Now assume $A$ is of rank ...
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### Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos
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### Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
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### Why if $aX+c=bX+d$ then $a=b$ and $c=d$?

There is theorem in linear algebra. I forgot it!! But I remember something from it. Can you please give me a reference? It is related to something like this. If I have two polynomials ...
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### Hermitian matrix the only diagonizable

During the last lecture one of my professors claimed that the hermitian matrix is the ONLY complex matrix which was diagonizable. This seems strange to mee (not to say a very very strong claim to ...
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### iterated dual vector spaces

Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-vector spaces belonging ...
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### Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
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### Some terminology and reference questions on singular values

Let $T: V \rightarrow W$ be an operator between to inner product spaces. Then singular values $s_1 \leq s_2 .... \leq s_n$ of $T$ are square roots of eigenvalues of $T^*T$ where $T^*$ is the conjugate ...