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Results tagged with linear-algebra
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user 41644
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
2
votes
Does $V \otimes V$ have non-trivial invariant subspaces under all unitaries of form $I \otim...
$\DeclareMathOperator\U{U}\DeclareMathOperator\Hom{Hom}$Write $d=\dim V$. Then $V$ is a $\U(V)$-representation. In the setting of the question, the space $V\otimes V$ is also a $\U(V)$-representation, …
- 4,969
3
votes
How many multiplications are needed to generate a matrix algebra?
One can receive some lower bounds on $N(d)$ from the theory of groups of central type. Let then $G$ be a finite group of order $d^2$. Assume that there exists a non-degenerate two cocycle $[\alpha]\in …
- 4,969
3
votes
Continuous linear combination of continuously varying vectors?
In case the vectors $e_1(t),e_2(t),e_3(t)$ form a basis for every $t$, you can solve Equation (E) and get $c_i(t)$ as rational functions in the coefficients of $e_i(t)$ with respect to some fixed basi …
- 4,969
8
votes
1
answer
214
views
Pair of square matrices related by traces formulas
Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{C}$. Assume that for every $k\geq 1$ it holds $tr(A^k) = tr(B^{2k-1})$. What can we say about the possible eigenvalues of $A$ and of $B$? How …
- 4,969
2
votes
Accepted
How to compute the joint spectrum?
Let us write $T=M_n(\mathbb{R})$. We have a polynomial map $f:\mathbb{R}^k\times T^k\to Hom_{\mathbb{R}}(T^k,T)$ which sends $((\lambda_i),(A_i))$ to the linear map $$(B_i)\mapsto \sum_i B_i(A_i-\lamb …
- 4,969
0
votes
Splitting subspaces and finite fields
The statement obviously holds for $W=S$. Take now any nonzero element $a\in K$ and take $W=S\cdot a$. Then since everything is commutative, and multiplication by $a$ is invertible, the statement is st …
- 4,969
5
votes
Accepted
Is an associative division algebra required for this phenomenon?
Consider the matrix $\sum_i a_iR_i$. One can show that it sends every vector of length 1 to a vector of length $\sqrt{\sum_i a_i^2}$. It follows that if the norm of $a=(a_i)$ is 1, then the matrix $\ …
- 4,969
10
votes
Accepted
minimum-maximum entries matrix
Let us write $$a_r=\frac{x_{r+1}}{x_r}$$ for $r=1\cdots n$.
We can then write the matrix $M(n)$ in the form
$$\begin{pmatrix} 1 & a_1 & a_1a_2& \cdots & a_1a_2\cdots a_{n-1} \\ a_1 & 1 & a_2& \cdot …
- 4,969