All Questions
6,287 questions
109
votes
19
answers
38k
views
Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
6
votes
3
answers
2k
views
Conjugate Gradient for a "slightly" singular system.
Suppose I have a symmetric $N \times N$ matrix A which has a one-dimensional Nullspace $N$. A is positive definite on $N^\bot$. In my case $N$ is the space of constant vectors (i.e. generated by ...
- 617
1
vote
2
answers
3k
views
matrices self-adjoint with respect to some inner product
Is it possible to give a nice characterization of matrices $A \in R^{n \times n}$ which are self-adjoint with respect to some inner product?
These matrices include all symmetric matrices (of course) ...
- 415
8
votes
2
answers
4k
views
Estimating the spectral radius of a matrix, noniteratively
Morris Marden's "Geometry of Polynomials" displays a number of formulae that allow one to estimate bounds on the largest root of a polynomial that do not involve actual rootfinding. Having been ...
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,200
1
vote
1
answer
201
views
How can I characterize the type of solution vector that comes out of a matrix?
Ax = b. I need a way to analyze a square matrix A to see if its solution vector x will ...
- 133
0
votes
1
answer
1k
views
For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?
Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...
- 135
5
votes
1
answer
263
views
Modeling free Lie algebras with matrix algebras
I am approximating some algebraic expressions of operators from a free Lie algebra. It is possible but messy to collect all independent operator objects of a given degree (same as grading?) that ...
- 731
34
votes
2
answers
4k
views
Symmetric powers and duals of vector bundles in char p
Suppose that $X$ is a smooth projective variety (eg $P^n$) and $E$ is a vector bundle (eg the tangent bundle). If the characteristic is zero, then taking symmetric powers "commutes" with taking duals:
...
- 798
14
votes
4
answers
6k
views
When is an algebra of commuting matrices (contained in one) generated by a single matrix?
Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...
- 30.1k
8
votes
5
answers
15k
views
Eigenvalues of A+B where A is symmetric positive definite and B is diagonal
If I have a symmetric positive definite matrix A and a diagonal matrix B, and I know the eigenvalues of both A and B (by iterative numerical computation in A's case and trivially for B), is there any ...
- 135
38
votes
10
answers
18k
views
Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
26
votes
3
answers
4k
views
How are these two ways of thinking about the cross product related?
I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner....
- 118k
6
votes
2
answers
1k
views
Proving equality of varieties by dimension counting
For finite sets $A$ and $B$, it is clear that $A \subseteq B$ and $|A| \geq |B|$ implies $A = B$. While an obvious fact, it can sometimes be a nice shortcut in proofs.
Analogously, if $V$ and $W$ are ...
- 2,071
66
votes
2
answers
8k
views
Geometric interpretation of characteristic polynomial
The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
- 2,071
1
vote
1
answer
434
views
Tori acting on vector spaces
Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now ...
- 11.2k
2
votes
3
answers
768
views
naming for the map $T = x \mapsto a x b$
Suppose $a,b$ are two matrices (arbitrary for now), and I have a function defined on a space of matrices, $T(x) = a x b$. This function is a linear and bounded transform on the a finite dimensional ...
- 155
5
votes
2
answers
457
views
fast merging of orthogonal bases
Given two matrices $U_1, U_2$, we can use QR factorization to find orthogonal basis for the subspace spanned by (columns of) $\begin{bmatrix}U_1,U_2\end{bmatrix}$.
Now this generally makes no use of ...
- 93
5
votes
2
answers
2k
views
Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
- 1,532
7
votes
1
answer
919
views
Association scheme on injective functions
This problem arises while studying the complexity of algorithms and I am quite unfamiliar with the subject.
Consider the set F of injective functions from {1..N} to {1..M}
we can define an ...
- 209
4
votes
2
answers
1k
views
Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian?
Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is ...
5
votes
2
answers
687
views
Dependence of trace norm on matrix size for smooth vs. random matrices.
Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...
- 846
9
votes
3
answers
1k
views
Vector spaces with natural bases
Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default ...
19
votes
17
answers
7k
views
Vector spaces without natural bases
Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural".
To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
13
votes
2
answers
3k
views
Left and right eigenvalues
A quaternionic matrix $A$ gives rise to a
function $\mathbb{H}^n \to \mathbb{H}^n$
given by $x \mapsto A \cdot x$. This is real linear,
but not complex- or quaternionic-linear
(in general) if we ...
- 12.5k
8
votes
3
answers
888
views
Techniques to bound products of upper triangular matrices and their inverses
Let $A_0, \dots, A_{n-1}$ be upper triangular matrices with ones on the diagonal. Let $B_{n-1}, \dots, B_0$ be of the same form.
I am interested in bounding
$$|| A_0 \dots A_{n-1} B_{n-1}^{-1} \dots ...
- 25.6k
3
votes
2
answers
3k
views
distributed incremental SVD
Hello all,
I need some theoretical pointers (formulas, articles, online links) on how to merge Singular Value Decompositions (SVD) of two matrices (two different sets of observations over the same ...
- 93
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
3
votes
1
answer
375
views
Connections between a polytope's symmetry group and the existence of periodic orbits
Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
- 269
4
votes
1
answer
1k
views
Stability of Conjugate Gradient Method
When dealing with spd matrices of relatively low condition number, how likely is it (and is it easily provable) that the conjugate gradient method will always be able to find the solution without ...
- 221
5
votes
2
answers
2k
views
Characteristic surface for systems of PDE
Despite the title, this is probably actually a question in linear algebra or algebraic geometry. Let me write the question(s) first, before I explain the background.
Problems
Let $h^{\mu\nu}_{ij}$ ...
- 39k
0
votes
3
answers
1k
views
Intuitions/connections/examples for "eigen-*"
There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...
- 5,935
34
votes
3
answers
22k
views
Singular values of matrix sums
This is a follow-up question to this one about eigenvalues of matrix sums. Suppose you have matrices $A$ and $B$, and know their singular values. What can you say about the singular values of $A+B$?
...
- 6,342
-1
votes
2
answers
806
views
The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
- 251
25
votes
3
answers
4k
views
Largest number of vectors with pairwise negative dot product
What is the largest $m$ such that there exist $v_1,\dots,v_m \in \mathbb{R}^n$ such that for all $i$ and $j$, $1\leq i< j\leq m$, we have $v_i \cdot v_j < 0$.
Also, the preview screen is not ...
user5810
2
votes
0
answers
1k
views
Good sources for linear algebra for convex optimization and graph analysis?
What are some good sources for linear algebra for convex optimization and graph analysis?
In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (...
- 2,383
28
votes
5
answers
4k
views
Does Smith normal form imply PID?
Let $R$ be a nonzero commutative ring with $1$, such that all finite matrices over $R$ have a Smith normal form. Does it follow that $R$ is a principal ideal domain?
If this fails, suppose we ...
user5810
12
votes
3
answers
3k
views
How to combine linear constraints on a matrix and its inverse?
Suppose there exists a $(n \times n)$ matrix $A$ that is real and invertible (nothing unusual or special about $A$). We do not know the entries of $A$. However, we do have linear constraints, some of ...
6
votes
5
answers
1k
views
A signature inequality?
Given two real symmetric matrices $A$ and $B$ of common square size $n$ with no strictly negative eigenvalues, can the symmetric matrix $AB+BA$ have strictly more than $n/2$ eigenvalues which are ...
- 17.9k
6
votes
2
answers
2k
views
Inner products and norms
Let $f:[n]\times [n] \rightarrow [0,1]$ be a function from pair of integers to the real interval [0.1]. I would like to find sets of complex vectors
$X= \{x_i\}$ and $Y=\{y_j\}$ satisfying $x_i\cdot ...
4
votes
1
answer
346
views
approximately linear functions -- more
Suppose $f,g$ are continuous functions from $\mathbb R$ to $\mathbb R$, with the property that
$$f(x)+f(y)=g(x+y)$$
for all $x,y$. Taking $x=y=z/2$ implies that $g(x)=2f(x/2)$ so that the above ...
- 123
1
vote
2
answers
1k
views
Assessing measurement accuracy and precision
I have been asked to assess the accuracy and precision of a new measurement method (Let's call it method B). This new method is compared to an existing one (A) that has its own specifications in ...
- 31
7
votes
1
answer
2k
views
approximately linear functions
i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies
$f(x-y)=f(x)-f(y)+const$
then it is necessarily linear.
are there any general ...
- 123
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 711
0
votes
2
answers
371
views
Quantum observables
Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B.
Question 1 : What is known about ...
10
votes
3
answers
6k
views
Solving a system of linear inequalities -- what is the dimension of the solution set?
It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...
- 7,857
12
votes
2
answers
8k
views
Is there a way to simplify block Cholesky decomposition if you already have decomposed the submatrices along the leading diagonal?
Let's say we have a block matrix $ M =\left( \begin{array}{ccc}
A & B\\
B^{*} & C \end{array} \right)$ where $M$ is positive definite. ($A$ and $C$ are also positive definite.)
There is a ...
9
votes
1
answer
1k
views
(Elementary?) combinatorial identity expressing binomial coefficients as an alternating sum over permutations.
Background
I came up with this while trying to find a sort of high-level exposition of the exterior algebra of a vector space. Let $V$ be a vector space of dimension $n$ over $\mathbb{C}$, and let $...
- 8,559
6
votes
2
answers
2k
views
Efficient approximation of a matrix and its inverse
Assume that $ A $ is a real $ n\times n $ matrix whose rows constitute an orthonormal basis of $ \mathbb R^n $.
Informal statement of question: Assume we want to approximate $ A $ by a rational ...
- 201