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Results tagged with matrices
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user 36688
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
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A quantity associated with an algebraic variete
Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that …
- 356
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2
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the spectrum of matrix with positive entries
It is well known that a matrix which all entries are positive real numbers, has a positive eigenvalue.(see algebraic topology, by Allen Hatcher). Now is the following generalization, true?
Let A b …
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The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$
Motivated by the following RG question we ask a related question as follows:
We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes GL(\math …
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The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra
Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X)
\otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ …
- 356
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459
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Is this a full rank matrix? [closed]
According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a_{ii} …
- 356
3
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1
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2k
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A norm description for singular matrices
For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property:
$A\in M_{n}(\mathbb{R})$ is singular if and only if $\para …
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373
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A line bundle over the manifold of singular matrices
According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$
So we define …
- 356
2
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1
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198
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Symplectic (contact) structure on $M_{n}(\mathbb{R})$
Under what contact structures on $M_{n}(\mathbb{R})$, the set of singular matrices is invariant under the flow of corresponding Reeb vector field? … Or a refined version: the set of $k$ rank matrices would be invariant under this flow. …
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Homotopy type of certain maps on complex grassmanian
$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M_{n}(\mathbb{C})\mid A=A^{*}=A …
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Is a pointwise " simple tensor" valued continuous map a tensor product of two continuous maps?
A matrix $A\in M_{4}(\mathbb{C})$ is called a simple tensor if $A=B\otimes C$ for two $2\times 2$ matrices $B,C$. …
- 356
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1
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368
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How to compute the joint spectrum?
Let $(A_{1},A_{2}, \ldots,A_{k})$ be $k$ matrices in $M_{n}(\mathbb{R})$. … the right ideal generated by $(A_{i} - \lambda_{i}I)_{i=1}^{k}$ is not the whole $M_{n}(\mathbb{R}).$
Moreover, motivated by Perron–Frobenius theorem we ask:
Assume that all enries of all matrices …
- 356
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Unitary factor in polar decompositions
proposition $2.4$ of this paper shows that for $n$ sufficiently large one can construct a $n \times n$ unitary matrix $U=-I_{2}\oplus U'$, which can be decomposed as the product of three positive matrices … This matrix, with distance $2$ from the identity matrix $I_{4}$, is the unitary factor of polar decomposition of $AB$ for two positive matrices $A,B$. …
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Recovering "$n$" from $M_n(\mathbb{C})$
Is there an example of an infinite-dimensional $C^*$-algebra $A$ which admits the following structure:
The $C^*$ algebra $A$ admits a faithful trace $tr$ such that the multiplication $m: A\otimes A \ …
- 356
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228
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A coalgebra structure on compact operators
Is there a coalgebra structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is compatible with the natural embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$ with $i_{n}(A)= A\oplus …
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