Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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1
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0answers
13 views

Diagonalization for sums of Hermitian matrices

I found an interesting question about diagonalizable matrices, Let $A,B\in \mathcal{M}_n(\mathbb{C})$ Hermitian, such that $AB\neq BA$. Is there exist $u\neq v$ complex numbers, such that $A+uB$ and ...
1
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0answers
49 views

Collecting terms with nested sums and combinatorics

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
1
vote
0answers
25 views

How do I prove a matrix A is self adjoint to an inner product? . [on hold]

In the source question B is an element of $M_n(R)$ and is a symmetic matrix such that $v^tBv>0$. Also $<.|.>$ is an inner product on $R^n$ called the $B$-inner product. we are asked to ...
2
votes
0answers
13 views

Efficiently factorize a KKT system with block diagonal upper corner

I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form: \begin{equation} A = \left[\begin{array}{c|c} ...
4
votes
0answers
171 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 ...
2
votes
1answer
41 views

Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here: Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
1
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0answers
35 views

Injectivity of a linear logistic transform

The motivation for this question has to do with neural networks, but it is essentially a purely mathematical question. Suppose you have a perceptron with one hidden layer, a bias, and a logistic ...
2
votes
0answers
96 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 $\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...
1
vote
2answers
37 views

Linear Programm with matrix [on hold]

Is there a name for problems like this min norm(Cx) Ax = b where C is a matrix and norm is the maximum norm. This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
3
votes
0answers
56 views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
0
votes
1answer
57 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, ...
0
votes
0answers
22 views

Finding a particular solution to the non-homogenous system [on hold]

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
3
votes
1answer
39 views

variation of the Lieb concavity theorem

A special case of the well known Lieb concavity theorem states that the following function is concave on positive operators A and B: $$ (A,B) \to \text{Tr} \{A^s X B^{1-s} X^\dagger \} $$ for $s \in ...
0
votes
0answers
61 views

proof non diagonalizable matrix is not an inner product [on hold]

Given $ A \in M_n(\Bbb C) $ and $ <x,y>_A = x^TA\overline y $ I need to proof that if A is non diagonalizable then $<.,.>_A$ is not an inner product. I thought about: Let A be non ...
6
votes
1answer
178 views

Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...
0
votes
0answers
49 views

Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system $$ x=A^{-1}b $$ This inverse problem is basically a least square problem with a rank-1 update. $$ ...
-3
votes
0answers
46 views

Math test and help [on hold]

Which statements are true for both translations and rotations? A. Transformed figures are congruent. B. Resulting line segments are parallel. C. Angle measures are preserved. D. Figure ...
3
votes
0answers
193 views

Find the Range of Function

What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$, $$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$ where $b_k=\prod_{j\neq k}(z_j-z_k)$ for $1\leq k\leq m$ ?
-2
votes
0answers
58 views

Hessian Matrix and Kronecker Product [on hold]

Given the following equation, $\Delta Y=J\Delta X+\frac{1}{2}H \Delta X \otimes \Delta X$ where $\Delta Y, \Delta X \in \mathbb{R}^{n}$, $J \in \mathbb{R}^{n \times n}$ is the Jacobian and $H \in ...
8
votes
2answers
614 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 ...
4
votes
3answers
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
1answer
115 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
1answer
60 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
1answer
114 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
0answers
21 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
2answers
328 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
1answer
242 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
0answers
61 views

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

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
0answers
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
0answers
86 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
1answer
43 views

How to find a matrix by its characteristic value and characteristic vectors? [closed]

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
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0answers
41 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 ...
0
votes
0answers
72 views

Eigenvalue of (0-1) matrix [closed]

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
0answers
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
0answers
285 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
0answers
201 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 $B_0,\dots,B_k$, ...
3
votes
1answer
118 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 ...
1
vote
0answers
34 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
6answers
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
0answers
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
0answers
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
2answers
244 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 ...
2
votes
4answers
335 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
2answers
430 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
0answers
31 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
1answer
85 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
1answer
250 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. ...
10
votes
1answer
209 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
0answers
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
0answers
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 ...