**35**

votes

**6**answers

2k views

### Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...

**27**

votes

**4**answers

31k 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 they are Hermitian and positive-definite?
I am investigating ...

**18**

votes

**2**answers

2k views

### Dimension of infinite product of vector spaces

This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...

**8**

votes

**2**answers

5k views

### Solving a quadratic matrix equation

This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? ...

**169**

votes

**22**answers

24k views

### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
This question ...

**31**

votes

**21**answers

10k 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 ...

**51**

votes

**4**answers

5k views

### explicit big linearly independent sets

In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or ...

**21**

votes

**1**answer

4k views

### Infinite Tensor Products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...

**7**

votes

**3**answers

1k views

### Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.
Background
A simple consequence of the singular value decomposition is that any vector ...

**16**

votes

**2**answers

1k views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**66**

votes

**8**answers

12k views

### Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\frac{1}{ad - bc} \begin{pmatrix} d & -b \\\ -c & a ...

**33**

votes

**17**answers

8k views

### Linear Algebra Proofs in Combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...

**54**

votes

**11**answers

6k views

### Why are matrices ubiquitous but hypermatrices rare?

I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...

**32**

votes

**7**answers

4k views

### Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find ...

**27**

votes

**7**answers

5k views

### “A gentleman never chooses a basis.”

Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.
Is there a gentlemanly way to prove that the natural map from V to V** is surjective if V is finite ...

**37**

votes

**4**answers

4k views

### Does the fact that this vector space is not isomorphic to its double-dual require choice?

Let $V$ denote the vector space of sequences of real numbers that are eventually 0, and let $W$ denote the vector space of sequences of real numbers. Given $w \in W$ and $v \in V$, we can take their ...

**25**

votes

**6**answers

20k views

### Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices.
I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.
Is this true in general for ...

**30**

votes

**2**answers

2k views

### The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio ...

**14**

votes

**3**answers

3k views

### Number of unique determinants for an NxN (0,1)-matrix.

I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore won't have a determinant. While it might also be ...

**7**

votes

**1**answer

759 views

### A spectral inequality for positive-definite matrices

Question. Given a positive-definite $n \times n$ matrix $A = (a_{ij})$ with eigenvalues
$$
\lambda_1 \leq \cdots \leq \lambda_n ,
$$
is there a sharp upper bound for the product $\lambda_2 \cdots ...

**22**

votes

**1**answer

1k views

### If $V$ is a vector space with a basis. $W\subseteq V$ has to have a basis too?

Suppose $V$ is a vector space, we say that $\mathcal B$ is a basis for $V$ if:
Every $v\in V$ can be written as a linear combination of elements of $\mathcal B$;
If $\sum\alpha_i b_i = 0$, where ...

**20**

votes

**4**answers

1k views

### Does Anyone Know Anything about the Determinant and/or Inverse of this Matrix?

The matrix I am inquiring about here is the $n \times n$ matrix where the entry $A_{ij}$ is $\frac{1}{(i+j-1)^2}$. The $2 \times 2$ matrix looks like
$$
\begin{pmatrix}
1 & 1/4 \\
1/4 & 1/9
...

**17**

votes

**8**answers

2k views

### Finitely presented sub-groups of GL(n,C)

Here are two questions about finitely generated and finitely presented groups (FP):
1) Is there an example of an FP group that does not admit a homomorphism to $GL(n,C)$ with trivial kernel for any ...

**14**

votes

**4**answers

965 views

### Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 ...

**5**

votes

**0**answers

257 views

### concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...

**4**

votes

**2**answers

903 views

### Iterated calculation of determinants

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...

**15**

votes

**1**answer

867 views

### Sizes of bases of vector spaces without the axiom of choice

Assuming the axiom of choice does not hold we have that there is a vector space without a basis. The situation can be, in some sense, worse. It is consistent that there are vector spaces that have two ...

**4**

votes

**1**answer

334 views

### The height of the Perron-Frobenius eigenvector

Does the height of a real symmetric matrix with non-negative entries control the height of its Perron-Frobenius eigenvector, under some reasonable definition of heights?
Just as an example of what ...

**10**

votes

**1**answer

193 views

### On Sampling rank $r$ matrices

Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $\{-d,\dots,-1,0,1\dots,d\}$ uniformly.
What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?
Is there a nice parametrization of ...

**6**

votes

**2**answers

158 views

### Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following:
For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...

**2**

votes

**0**answers

410 views

### On Eigenspace of a Bundle Map which is the horizontal part of a complex structure on $TM$

Let $(M^{n+m},g)$ be a Riemannian manifold and let $\mathcal{H}(TM) \subseteq TTM$ be the horizontal space associated to the Levi-Civita connection of $g$. Let $\bar{J} : TTM \longrightarrow ...

**1**

vote

**5**answers

308 views

### About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...

**12**

votes

**5**answers

900 views

### Does k(X) have a k-basis for every set X, without AC?

This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?.
For any field $k$, the field $k(x)$ of rational functions in one variable has an ...

**5**

votes

**1**answer

386 views

### “Orthogonal complement” in $\mathbb{Z}_q^n$

Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}_q^n$ with addition componentwise where $\mathbb{Z}_q$ is the integers mod $q$. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w ...

**4**

votes

**2**answers

764 views

### How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...

**3**

votes

**1**answer

204 views

### Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...

**2**

votes

**3**answers

687 views

### Continuous change of basis (and on the definition of determinant) [closed]

Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...

**1**

vote

**1**answer

209 views

### A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function.
What matrices belongs to $S$, precisely?
Let ...

**65**

votes

**3**answers

12k views

### Slick proof?: A vector space has the same dimension as its dual if and only if it is finite dimensional

A very important theorem in linear algebra that is rarely taught is:
A vector space has the same dimension as its dual if and only if it is finite dimensional.
I have seen a total of one proof ...

**60**

votes

**19**answers

23k 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. ...

**40**

votes

**18**answers

37k views

### Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...

**38**

votes

**19**answers

8k views

### Wonderful applications of the Vandermonde determinant

This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...

**54**

votes

**11**answers

6k views

### Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...

**53**

votes

**3**answers

3k views

### Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...

**31**

votes

**4**answers

3k views

### Using linear algebra to classify vector bundles over P^1

There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let ...

**49**

votes

**7**answers

4k views

### How to prove this determinant is positive?

Given the matrices $ A_i=
\biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)
$, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...

**32**

votes

**9**answers

8k views

### Can a vector space over an infinite field be a finite union of proper subspaces?

Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces?
If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are ...

**33**

votes

**10**answers

8k views

### real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?

**31**

votes

**3**answers

3k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**27**

votes

**9**answers

12k 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 ...