All Questions
3,680 questions
1
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876
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Matrix logarithms are not unique
In my ODE class, we proved that if $\exp(L) = \exp(L')$ then the eigenvalues are congruent mod $2 \pi i$. Here, $L$ and $L'$ are two $n \times n$ matrices. I wanted to know if something more precise ...
- 22.8k
5
votes
1
answer
2k
views
Self-similar matrices? [closed]
Does anyone know anything about self-similar (infinite) matrices, with more or less fractal(-like) structure and admitting meaningful matrix-algebra operations?
- 101
9
votes
4
answers
1k
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presentation for GL(n,K)
let $K$ be a field, $n \geq 1$. denote $E_{i,j}$ the elementary matrix having $1$ on the diagonale and in the entry $(i,j)$, and $E_i(a)$ the elementary matrix $diag(1,...,a,...,1)$. you know that $...
- 63.1k
1
vote
2
answers
922
views
Extremum under variations of a traceless matrix
Sorry for my precedent tentative, I was a little hasty:
Ok, I think I'd better put the original problem:
I have an action of three fields: $A$ which is the spin-connection, $B$ an skew-symmetric 2-...
- 733
8
votes
4
answers
8k
views
Is there an existing name for "piecewise vector multiplication"
Given two vectors of size $n$
$$u = [u_1, u_2, u_3, ..., u_n ] $$
and
$$v = [v_1, v_2, v_3, ..., v_n ] $$
What is the name of the operation "$u ? v$" such that the result is a vector of size $n$ of ...
- 339
14
votes
3
answers
1k
views
"Conjugacy rank" of two matrices over field extension
I have posted this elsewhere and got only a partial reply. I don't know whether this qualifies the question for an open-problem tag; if it does, please anyone insert it.
Let $L$ be a field, and $K$ a ...
- 33.8k
29
votes
12
answers
6k
views
When does 'positive' imply 'sum of squares'?
Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...
2
votes
2
answers
2k
views
The application of Lanczos Algorithm on sparse matrix
I am looking for suitable algorithm to compute the eigenvalues and eigenvectors of a matrix. My matrix is sparse ( think of Finite Element Matrix), and it is very, very big ( think of hundreds of ...
- 381
2
votes
1
answer
651
views
Splitting matrix of rank one
Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
rank A=1 ↔ all 2 x ...
4
votes
5
answers
5k
views
conjugate gradient iteration
I'm having problems understanding why the conjugate gradient method breaks down for singular matrices. I've read a good introduction to intuitively understanding the CG method through visualizing the ...
- 51
0
votes
2
answers
119
views
Properties of adjacent submatrixes [closed]
Hi!
I've encountered a matrix problem when designing an algorithm, which I cannot seem to figure out. I have a (square) matrix with the following properties:
j<k → aij<aik, aji<aki
aij&...
- 101
5
votes
5
answers
5k
views
Notions of Matrix Differentiation
There are a few standard notions of matrix derivatives, e.g.
If f is a function defined on the entries of a matrix A, then one can talk about the matrix of partial derivatives of f.
If the entries of ...
- 590
2
votes
4
answers
3k
views
Symmetrical Presentation of 4-Dimensional Rotation Matrix
This question is not urgent; just a matter of curiosity...
It is relatively easy to generate an arbitrary 3D or even 4D rotation matrix using conjugation (i.e. YXY−1) of orthogonal rotations. I ...
- 524
8
votes
2
answers
2k
views
Hermitian matrices with prescribed number of positive and negative eigenvalues
Let $H$ be a linear subspace of the space of Hermitian $n\times n$ matrices. Is there a good characterization of those $H$ such that every $A\in H$ has at least $k$ positive and $k$ negative ...
- 971
31
votes
10
answers
9k
views
When to pick a basis?
Picking a specific basis is often looked upon with disdain when making statements that are about basis independent quantities. For example, one might define the trace of a matrix to be the sum of the ...
- 2,649
1
vote
2
answers
5k
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Elliptical rotation matrix [closed]
We can rotate a point 'circularly' about an arbitrary axis:
the equation is here, but this site doesn't trust me enough yet to post an image.,
But as we walk theta 0 -> 2PI this takes the point ...
- 133
1
vote
4
answers
385
views
Is it that only with normal matrices, the transition matrix to its [del: inherent] [ins: own] basis is unitary?
Does this even make sense what I translated into english?
PS. I am probably gonna delete this question eventually
- 13
91
votes
5
answers
124k
views
Eigenvalues of matrix sums
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am ...
- 1,013
2
votes
1
answer
978
views
What is the comultiplication of a matrix frobenius algebra?
One of the easiest examples I can think of for frobenius algebras is a plain ol' matrix algebra with tr : V → k as the co-unit (or equivalently, tr(a⋅b) as the frobenius form). This is ...
- 1,872
4
votes
4
answers
1k
views
Distribution of 1-norm for Gaussian Unitary Ensemble
Suppose I uniformly sample matrices X from the Gaussian Unitary Ensemble (GUE) with variance \sigma^2. Consider the Ky-Fan d norm, i.e. the sum of the singular values, of X. Let's call this Z=||X||...
- 2,649
19
votes
4
answers
2k
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Variation on a matrix game
The original problem appeared on last year's Putnam exam:
"Alan and Barbara play a game in which they take turns filling entries of an initially empty 2008×2008 array. Alan plays first. At each turn, ...
- 1,975
8
votes
1
answer
948
views
Surjective maps given by words, redux
I asked some time ago:
Let $w(X,Y)$ be a word in $X$ and $Y$ (i.e., an element in the free group on $X$ and $Y$). Let the variables $x$ and $y$ now range among elements of $SL_n(K)$, $K$ an ...
- 20.2k
22
votes
2
answers
14k
views
Infinite matrices and the concept of "determinant"
Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
- 1,861
9
votes
1
answer
643
views
Determinant of a pullback diagram
Suppose that X and Y are finite sets and that f : X → Y is an arbitrary map. Let PB denote the pullback of f with itself (in the category of sets) as displayed by the commutative diagram
PB &...
- 2,600
14
votes
2
answers
6k
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What is the constant of the Coppersmith-Winograd matrix multiplication algorithm
Or at least it's order of magnitude.
I've only ever heard it described as "huge", and a google search turned up nothing.
Also, given that the Strassen algorithm has a significantly greater constant ...
- 2,383
26
votes
8
answers
4k
views
Euclidean volume of the unit ball of matrices under the matrix norm
The matrix norm for an $n$-by-$n$ matrix $A$ is defined as $$|A| := \max_{|x|=1} |Ax|$$ where the vector norm is the usual Euclidean one. This is also called the induced (matrix) norm, the operator ...
- 365
11
votes
1
answer
410
views
An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
40
votes
6
answers
6k
views
Linear transformation that preserves the determinant
It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on $n\times n$ matrices with complex coefficients that preserves the determinant. Then there exists ...
- 511
7
votes
6
answers
1k
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Does the space of $n \times n$, positive-definite, self-adjoint, real matrices have a better name?
This is also the space of real, symmetric bilinear forms in $\Bbb R^n$.
- 8,502
9
votes
6
answers
8k
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How to approximate a solution to a matrix equation? [closed]
Suppose a matrix equation $Ax = b$ has no solution ($b$ is not in the column space of $A$)
How can I find a vector $x^\prime$ so that $Ax^\prime$ is the closest possible vector to $b$?
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