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Results tagged with matrix-analysis
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user 36688
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
1
vote
3
answers
195
views
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
1
vote
1
answer
459
views
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
5
votes
1
answer
186
views
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 $B,C$.
Assume that $X$ is a Hausdorff topological space.Assume that $f:X\to M_{4}(\m …
- 356
7
votes
Accepted
A matrix norm inequality II
No the norm of the left side can be very large.
For example $\left\| X^{-1}AXB \right\| $ is an unbounded function in $(x,y)$ where $A,X,B$ are the following matrices:
Put $A=B= \begi …
- 356
2
votes
Unitary factor in polar decompositions
I think that the part $(a)$ of 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 decomp …
- 356
2
votes
1
answer
261
views
A certain subset of general linear group
Motivated by the concept of diagonally dominated matrices we consider the space $S$ of all complex $n\times n$ matrices with $|a_{ii}|>\sum_{j\neq i} |a_{ij}|$, for every $i$. Every elemen …
- 356
0
votes
1
answer
147
views
Is it a sufficient condition for linearity?
Edit: According to the comment by LSpice we come back to the initial version of this question
Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the der …
- 356
2
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ ma...
It is sufficient to prove the statement in the question only for matrices in the form $[1]_{1\times 1}\oplus [0]_{k}$, that is the rank-1 projections or $$\begin{pmatrix} 0&\lambda\\ -\lambda …
- 356
3
votes
1
answer
171
views
Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$
Every matrix $A\in M_4(\mathbb{R})$
can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$.
What is the least uniform upper bound $M$ for such $n(A) …
- 356
0
votes
1
answer
199
views
Riemannian metrics on matrix space for which the restriction of trace function to each compl...
Edit: According to comment by Leo Monsaingeon I revise my question:
Is there a Riemannian metric on $M_n(\mathbb{R})$ for which the function $trace$ is a bounded function on every complete(whole …
- 356
3
votes
1
answer
538
views
The Matrix form of the Van der Pol equation
Motivated by the classical Van der Pol equation which has a unique periodic attractor, we consider the following differential equation on $M_{2}(\mathbb{R})\times M_{2}(\mathbb{R}):$
$$(*)\;\;\;\b …
- 356
6
votes
1
answer
751
views
Is every real matrix conjugate to a semi antisymmetric matrix?
Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with th …
- 356
3
votes
Accepted
Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ ...
One can not say that $\mathcal E$ is always connected.
For $n=3, $(and similarly $n>3$) let $A$ be a matrix with $a_{i1}=1,\quad \forall i \in \{1,2,\ldots,n\} $
Then $\mathcal E$ conta …
- 356
29
votes
1
answer
3k
views
Is there an explicit formula for the hessian of "Determinant"?
Let $f: G= \mbox{GL}(n,\mathbb{R}) \to \mathbb{R}$ be the determinant function.
Then $\mbox{Hess} (f)$ is a two linear map on $M_{n}(\mathbb{R})\simeq T_{e}(G)$ where $e$ is the neutral element of $G$ …
- 356