All Questions
67 questions
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
19
votes
4
answers
7k
views
Sherman-Morrison type formula for Moore-Penrose pseudoinverse
Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$.
Then the Sherman-...
0
votes
1
answer
328
views
Number of minimal left ideals
Is there any way to compute the number of minimal left ideals of $M_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?
user39204
7
votes
1
answer
654
views
Existence of a generalized matrix inverse over an arbitrary field?
Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices
(also called pseudo-inverse $A^{\dagger}$ of $A$, ...
- 12.1k
3
votes
0
answers
337
views
when upper triangular matrix modulo prime ideals implies upper triangular?
Let $E/ \mathbb{Q}_p$ be a finite extension, let $\mathcal{O}$ be the ring of integers of $E$. Let $A$ be a reduced noetherian local complete $\mathcal{O}$-algebra with the maximal ideal $\mathfrak{m}$...
13
votes
1
answer
2k
views
Number of idempotent $n\times n$ matrices over $\mathbb{Z}/m\mathbb{Z}$?
Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m:=\mathbb{Z}/m\mathbb{Z}$ ?
The number of idempotent matrices over a finite field is well-known and ...
user31416
5
votes
4
answers
8k
views
Proving a determinant = 0
The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...
- 51
26
votes
2
answers
3k
views
Singular values of sequence of growing matrices
I asked this question on math.stackexchange and haven't received an answer in two weeks, so I'm repeating it here.
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \cr
1/2 & 0 &...
- 656
3
votes
1
answer
473
views
Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)...
Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...
- 24.9k
5
votes
1
answer
1k
views
Algebra - Decomposition of a matrix polynomial
Dear All,
This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer.
What is known about a possible ...
- 51
9
votes
3
answers
2k
views
On similar matrices and polynomial matrices
I'm teaching linear algebra and I'm encountering this theorem:
two matrices $A$ and B are similar iff $tI - A$ and $tI - B$ are equivalent (as polynomial matrices), where $I$ is the unit matrix.
The ...
- 357
3
votes
1
answer
303
views
ABA-product of matrices and length of chains of principal inner ideals
Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we ...
- 6,614
3
votes
1
answer
218
views
decompositions of matrices over $\mathbb{Q}$
Given a matrix $A\in GL_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL_n(\mathbb{Z}[1/p])$ and $C\in GL_{n}(\mathbb{Z}_{(p)})$, where $ \mathbb{Z_p}$ denotes the ...
- 11.1k
22
votes
1
answer
33k
views
vector to diagonal matrix [closed]
For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector.
Is there a simple way to write this transformation ...
- 247
2
votes
1
answer
3k
views
Is it possible to decompose a symmetric, positive definite matrix in this way?
Let $\Sigma$ be a symmetric positive definite matrix. Then the Cholesky decomposition gives us $\Sigma=LL'$ where $L$ is lower triangular and unique.
Under what conditions (if any) does there exist ...
- 269
2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
- 1,422
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427