11
votes
Accepted
What is the elementary proof of Weil's polynomial theorem of decomposition?
If $P(x,y),Q(x,y)$ are relatively prime, then so are the one-variable polynomials $p(x)=P(x,1),q(x)=Q(x,1)$ (since we can homogenize any common factor of $p,q$ to a common factor of $P,Q$). It follows ...
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7
votes
What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?
I don't know how to characterize such morphisms, which I think is your first question. However, this can certainly happen, even if $f$ is not smooth. (By the way, your comment about absence of local ...
- 35.2k
5
votes
Accepted
What is this matrix decomposition called and does it exist always?
No, you can't always find two such matrices.
Let $n \gt 2$. Choose a $2$-dimensional subspace $V$ of $\mathbb{Q}^n$ that does not intersect the positive orthant except at $\vec 0$. Let $M$ be a ...
- 28k
4
votes
Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$
Yes, there is, using exterior product as a functor: Let $e_i$ be the basis of $\mathbb R^n$. Then
\begin{align}
\det(A.D.A^\top)e_1\wedge\dots\wedge e_n &= \Lambda^n(A.D.A^\top)(e_1\wedge\dots\...
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4
votes
Singular Value decomposition of huge dimensional matrix
This problem of updating the singular value decomposition of a matrix upon repeatedly appending a row or a column to the matrix is discussed in A stable and fast algorithm for updating the singular ...
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4
votes
Accepted
Decomposition of Henstock-Kurzweil-integrable functions
The answer is negative. Consider the space $\Delta'=\Delta'([a,b])$ of functions, which are classical derivatives of functions $[a,b]\to\mathbb R$. Assume the claim was true. Then any HK-integrable $f:...
- 231
4
votes
Accepted
The closure of cyclic modules under direct sums and direct summands
The introduction of
Warfield, R.B.jun, Rings whose modules have nice decompositions, Math. Z. 125, 187-192 (1972). ZBL0218.13012.
says
It is also shown that if $R$ is a commutative ring and there ...
- 35.2k
2
votes
Is there an elementary proof of the polar factorization theorem for vector-valued function?
Here's an elementary type argument I learned, based on unique splitting $T=S+A$ of linear operators $T: \mathbb{R}^n \to \mathbb{R}^n$.
(i) every linear operator $T$ can be uniquely decomposed $T=S+A$ ...
- 2,274
2
votes
Accepted
Connections between eigenvectors after matrix multiplication
$U_s$ is not recoverable from $H$ and $U_A$.
Consider the following examples, one each for $M < N, M > N, M = N$. In each example, there are 2 different $R$'s, $R1$ and $R2$, having different $...
- 1,517
2
votes
Accepted
Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them
Since $\sigma_1^2=\lambda_+$ and $\sigma_2^2=\lambda_-$ are the two eigenvalues of the symmetric matrix product $MM^t$, we have $\lambda_++\lambda_-={\rm tr}\,MM^t=\|m_1\|^2+\|m_2\|^2$. Hence we may ...
- 188k
2
votes
Accepted
Decomposition of direct image of a smooth morphism, Deligne's theorem, motives
We can give many counterexamples to semisimplicity in positive characteristic using the observation that if the canonical morphism $k_Y\rightarrow f_*k_X$ in $D^b_c(Y,k)$ is split, then the induced ...
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2
votes
Accepted
What can be said about a projective morphisms that admit decomposition theorem like smooth morphisms?
Here is an example where $f$ is not smooth but $Rf_* \mathbb{C}$ behaves as if it were:
Let $X$ be a hyperelliptic surface and $f$ the natural morphism to $Y \cong\mathbb{P}^1$. All reduced fibres of $...
2
votes
Accepted
Is it possible to prove the Jordan decomposition starting from Schur's decomposition?
I am not sure I get what you mean by "brought to Jordan form", but if you don't consider the structure of $D$ while changing basis for $N$ then it won't work. Example:
$$
U =
\begin{bmatrix}...
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1
vote
Adding valid cuts for integer feasibility problem under Benders decomposition framework?
Yes. Search for combinatorial Benders decomposition or logic-based Benders decomposition. In particular, Benders feasibility cuts for a binary master problem are “no-good” cuts of the form $$\sum_{j\...
- 5,429
1
vote
Decomposition theorem over more general base schemes
If $X$ is any excellent scheme with equicodimensional irreducible components, there is a perverse t-structure on $D^b_c(X,\mathbf{Q}_\ell)$ with all expected properties ($\ell$ any prime invertible on ...
- 13.1k
1
vote
Accepted
Generate a two-variable polynomial from its "roots
There a number of very well researched techniques that can be used to solve your problem in a practical sense. Most of these have come from computer graphics and computer vision where taking a set of ...
- 4,862
1
vote
Find a minimum set of paths that cover all pairs of dependent vertices
An answer via hypergraph theory
There exists an obvious reduction to the minimum edge cover problem in hypergraph theory (also known as the set cover problem). I will describe this reduction.
A ...
- 6,051
1
vote
A Krull-Schmidt Theorem for Lie groups?
For ease of writing, I'll say a compact connected Lie group $G$ has the Krull–Schmidt property if, up to order, it has a unique decomposition as a direct product where each factor itself cannot be ...
- 6,546
1
vote
Examples for Decomposition Theorem
A nice example of the decomposition theorem for a proper morphism of smooth varieties is the blowup of the points $p_i$ lying at the intersection of two plane curves; that is, let $f, g \in \mathbb{C}[...
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