**8**

votes

**2**answers

603 views

### How to transform matrix to this form by unitary transformation?

Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e.
$$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$
Then is ...

**0**

votes

**0**answers

8 views

### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...

**4**

votes

**3**answers

1k views

### How many commuting nilpotent matrices are there?

To be precise, fix $n$, fix a field $k$.
What is the maximal dimension of a subspace of the vector space of all $n\times n$ matrices formed by commutative nilpotent matrices? By commutative I mean ...

**3**

votes

**1**answer

110 views

### Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that
$\|A\| < x$ and ...

**3**

votes

**1**answer

55 views

### submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...

**1**

vote

**1**answer

110 views

### Subgradient of Minimum Eigenvalue

Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...

**0**

votes

**0**answers

17 views

### Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem
\begin{align}
\lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\
&r_i\geq ...

**9**

votes

**2**answers

307 views

### What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...

**5**

votes

**1**answer

240 views

### Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...

**-5**

votes

**0**answers

58 views

### Prove determinant of nxn matrix is (a+(n-1)b)(a-b)^(n-1)? [on hold]

Prove det(mat) is (a+(n-1)b)(a-b)^n-1 where matrix is nxn matrix with a's on diagonal and all other elements b, off diagonal?
For example, suppose matrix with diagonal composed solely of a's. All ...

**-1**

votes

**0**answers

11 views

### regu tools l_curve regularization stanford ee 263 [migrated]

I am trying to solve one of the famous stanford EE263 problems, which gives me matrix A representing blurring of an image and y, representing the blurred image. For that I have been trying to use ...

**3**

votes

**0**answers

77 views

### Infinite series of determinants

I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...

**-2**

votes

**1**answer

41 views

### How to find a matrix by its characteristic value and characteristic vectors? [on hold]

Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen vlaues) and characteristic vectors (eigen vectors). But my qustion is why cant we ...

**1**

vote

**0**answers

37 views

### Lanczos algorithm with thick restart on a dynamic matrix

currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries from time to time. The problem should be solved using an ...

**3**

votes

**0**answers

125 views

### Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...

**0**

votes

**0**answers

71 views

### Eigenvalue of (0-1) matrix [on hold]

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10)
The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...

**3**

votes

**0**answers

68 views

### What do we know about the generalized eigenvalue problem involving a projector?

Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.
Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem
...

**14**

votes

**0**answers

280 views

### a naive question about the second dual of a vector space

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor
$$
V\mapsto V^{**}/V
$$
of the category of $K$-vector spaces?
I asked a related question on Mathematics Stackexchange, but ...

**8**

votes

**0**answers

195 views

### Lift chain complex from $F_2$ to $Z$

We start with a finite dimensional chain complex over $F_2$, equipped with a basis. That is, we have finitely many finite dimensional $F_2$-vector spaces $C_0,\dots,C_k$ with bases $B_0,\dots,B_k$, ...

**2**

votes

**1**answer

109 views

### About partial uniqueness of SVD

In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let ...

**0**

votes

**0**answers

31 views

### Bounding Rayleigh quotioent for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...

**50**

votes

**6**answers

4k views

### Existence of a zero-sum subset

Some time ago I heard this question and tried playing around with it. I've never succeeded to making actual progress. Here it goes:
Given a finite (nonempty) set of real numbers, $S=\{a_1,a_2,\dots, ...

**1**

vote

**0**answers

43 views

### For Finite Dual when is $(A \otimes A)^o = A^o \otimes A^0$?

Let $A$ be any $k$-algebra. The finite dual or restricted dual of $A$ is
$$
A^o = \{f \in A^* ~ | ~ f(I)= 0, \text{ for some ideal } I \subseteq A, \text{ such that } \text{dim}_k(A/I) < \infty\}.
...

**2**

votes

**0**answers

35 views

### successive schur complements

If I have a large (e.g. 6000x6000), sparse, positive definite matrix $M$ (which may have individual entries everywhere, but most non-zero entries are on / around the diagional).
Divide $M$ into blocks ...

**5**

votes

**2**answers

234 views

### How to check whether a matrix is completely positive or not?

The definition:
cone of completely positive matrices
$\mathcal{C}=\{\sum_{i=1}^kx_ix_i^T:x_i\in\mathbb{R}^n_+\ for \ i=1,2,...,k\}$.
I just don't knwo how to check whether a matrix belongs to ...

**0**

votes

**0**answers

41 views

### Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then
$\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{ii}|-\sum_{j \neq i}|a_{ij}|}$, where the ...

**2**

votes

**4**answers

334 views

### Efficient algorithm for finding the minima of a piecewise linear function

Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two ...

**-1**

votes

**2**answers

429 views

### Approximating a subspace by sampling a base without replacement

Let $X$ be a $p \times n$ matrix, with $p > n$. Now, suppose I sample $m < n$ columns from $X$ at random, without replacement. I would like to characterize the distance between the subspace ...

**3**

votes

**0**answers

29 views

### Simultaneous Tridiagonalization of a given set of hermitian matrices?

I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?

**7**

votes

**1**answer

84 views

### Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity

I'm currently trying to get familiar with the Jordan normal form for matrices; and after some example I ask for the possible Jordan-form for the Carleman matrix for the function $f(x) = \sin(x)$ when ...

**0**

votes

**1**answer

247 views

### eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...

**-1**

votes

**0**answers

42 views

### Feature relationship based class separability [closed]

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description.
I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...

**10**

votes

**1**answer

208 views

### Hölder's inequality for matrices

I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But ...

**7**

votes

**0**answers

84 views

### When is a product of hyperbolic matrices hyperbolic?

Suppose $A_1,\ldots,A_n$ is a sequence of $2 \times 2$ complex matrices such that $| \det(A_j) | =1$ and $ | \mathrm{tr}(A_j) | > 2 $ for each $j$. What kinds of reasonable restrictions can one ...

**1**

vote

**0**answers

85 views

### Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...

**0**

votes

**0**answers

74 views

### Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example,
$(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...

**5**

votes

**4**answers

261 views

### NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...

**0**

votes

**0**answers

33 views

### Compare full-rank probabilities of products of random matrices

Consider two matrices $C_1=A\times B_1$ and $C_2=A\times B_2$, where $A\in\mathbb{F}_q^{N\times K}$, $B_1\in\mathbb{F}_q^{K\times M}$ and $B_2\in\mathbb{F}_2^{K\times M}$; $M\leq N\leq K$.
It is ...

**0**

votes

**1**answer

118 views

### Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.
I'm ...

**-2**

votes

**0**answers

19 views

### how to complete arbitrary basis knowing 2 orthonormal vectors of Rd (d > 2) [migrated]

In a paper the following statement is used: "To construct the matrix B, complete the vectors (y, x) to an arbitrary basis of Rd and then apply the Gram-Schmidt orthonormalisation".
assume we know x ...

**3**

votes

**0**answers

135 views

### Is there such a matrix in $SO(n)$?

Given two $n$ dimensional positive definite matrices $A', B'$, is there a matrix $O \in SO(n)$ such that $A=O A', B=O B'$ and
$$
\frac{A_{ij}}{\sqrt{A_{ii}A_{jj}}} = ...

**3**

votes

**3**answers

192 views

### Square root of a complex matrix commuting with a given one

Assume two commuting $n\times n$ complex matrices $A$ and $B$ are given. Then it is in general false that if $C$ is a square root of $A$, i.e., if $C^2=A$, then $C$ commutes with $B$ (the simplest ...

**8**

votes

**2**answers

684 views

### On the Positive Definiteness of a Linear Combination of Matrices

In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated.
QUESTION:
Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...

**34**

votes

**1**answer

2k 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 ...

**3**

votes

**1**answer

179 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 ...

**3**

votes

**1**answer

74 views

### Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...

**-4**

votes

**0**answers

20 views

### Relationship for cosine of angle [migrated]

If $x$ is the cosine of the angle between the vectors $a$ and $b$, $y$ is the cosine of the angle between the vectors $a$ and $p$, and $z$ is the cosine of the angle between the vectors $b$ and $p$, ...

**4**

votes

**1**answer

1k views

### Upper bound on largest eigenvalue of a real symmetric n*n matrix with all main diagonal >0, everywhere else <=0

Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...

**3**

votes

**1**answer

142 views

### A hyperplane inside another one

Let D be a divison ring, let V be a left vector space of over D,
possibly infinite dimensional, and let F be the prime field of D.
Is it true that every F-hyperplane of V contains a D-hyperplane of ...

**0**

votes

**1**answer

60 views

### dual space of the quotient space of some locally convex topological space

I would like to a classical result about dual space. Let $E$ be a locally convex space and $F$ its closed linear subspace. If $E^{\ast}$ is the dual space of $E$, could some one affirm me that the ...