# Tagged Questions

Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to [tag:linear-algebra]). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur ...

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### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
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### Second-order perturbation expansion for singular value decomposition

Let $A = U\Sigma V^T$ be the singular value decomposition (SVD) of a $n\times m$ matrix $A$. Let $\tilde{A} = A + \epsilon P$ be a perburbation of $A$. It is possible, using tools from Matrix ...
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### Why does the objectivity rule out the convexity?

In the famous work "Ball J M. Convexity conditions and existence theorems in nonlinear elasticity[J]. Archive for rational mechanics and Analysis, 1976, 63(4): 337-403", it was mentioned that the ...
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### Largest eigenvalues distribution of tridiagonal symmetric random matrix

I would like to find the largest eigenvalue distribution of the following tridiagonal symmetric random matrix in an analytic way. All the ${\lambda}_i$ are distributed the same way with chi-square (...
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### Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent $$\text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j }$$ of a 3-tensor $W_{j,k,l}$ of ...
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I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements): $$\left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a &... 0answers 55 views ### max min of ratio of quadratic forms Consider the optimization over two vectors x and y$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$for two positive definite matrices A and B. This problem can be ... 0answers 74 views ### The state-transition-matrix of a physical system, Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ... 0answers 54 views ### inverse of asymptotic Toeplitz matrix with band limited associated function I am reviewing a controversial paper, and the main result, a revolution within my field, comes down to whether or not the following is true. I strongly believe it is not, but would need confirmation. ... 0answers 100 views ### Boundary of pseudospectra Suppose: B_i \in \mathbb{C}^{n \times n}, 0<w_i\in \mathbb{R} (i = 0,1,2,\ldots,m) {\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0 is a matrix polynomial, and x  is a complex ... 0answers 40 views ### Limits on a parameter \alpha to get positive definite matrix Given positive semi-definite n\times n matrices B_k, how would I go about getting the limits on \alpha_k such that the expression \mathbb{I}-\sum_{k=1}^{m-1}\alpha_kB_k \end{... 0answers 77 views ### Eigenvalues of a partitioned self-adjoint matrix This is a repost of the same question on MSE (with no reply/comment): http://math.stackexchange.com/questions/1454314/eigenvalues-of-a-partitioned-self-adjoint-matrix I would be grateful just for a ... 0answers 92 views ### Is my particular finite dimension Toeplitz matrix always strictly positive? Let H(\omega), \; -\pi \leq \omega \leq \pi be a real-valued function with a continuous band of zeros, that is (for simplicity) H(\omega)=0, \; |\omega|\geq \beta \pi. Define a sequence of banded ... 0answers 146 views ### Bounding the largest Singular value D is a n \times n diagonal matrix whose diagonal entries lies in (0,1]. B is any n \times n n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: (D+B)^{-1}D^2(... 0answers 43 views ### Constructing pointwise-or independent binary matrices Let M be an m \times n matrix such that every M_{ij} \in \{0,1\}. We say that M is `\lor-irreducible' if (i) no row is a pointwise-or of other rows, and (ii) no column is a pointwise-or of ... 0answers 161 views ### Asymptotic expansion square root matrix I am looking for an asymptotic expansion for \underline\gamma which is the "square root" matrix of a symmetric p\times p matrix \gamma. Here \underline\gamma is assumed to be symmetric, e.g. ... 0answers 101 views ### Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil Given a Hermitian indefinite pencil (A-\lambda B) where both A=A^H and B=B^H \in \mathbb{C}^{n\times n} are possibly indefinite, it is straightforward to show that the eigenvalues are either ... 0answers 211 views ### Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix I'm trying to find upper boundaries on the smallest Eigenvalue \lambda_1 of L + E, where L is a standard Laplacian of an unweighted digraph, with \lambda_1(L) = 0 and E \in \{0,1\}^{n \times ... 0answers 68 views ### Possible diagonal values of a product of matrices with some specific characteristics Hello all, This is a question that might or might not be related to my previous one. Imagine you have two matrices: Matrix \mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M} where M\... 0answers 94 views ### Characterizing the singular values of a matrix with structure Suppose we have a function from \mathbb{R}^2\to\mathbb{C},$$f(x,y) = e^{\imath\pi x g(y)}$$where g(y) is periodic in y\in[-T, T),\ T<\infty (e.g., a sinusoid) and 0\leq x < \infty ... 0answers 158 views ### Ring-theoretic version of a matrix problem Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of k orthogonal matrices. They proved that it is ... 0answers 134 views ### bounds on the entries of an inverse circulant matrix Suppose that C is a (real) circulant invertible matrix defined by a vector d. Then C^{-1} is also a circulant defined by some vector f. There exists a standard formula that expresses the ... 0answers 63 views ### sharper interlacing The usual interlacing inequalities say that if M is a Hermitian n \times n matrix and \hat{M} is a principal submatrix of order n-1, then \lambda_{\min}(M) \leq \lambda_{\min}(\hat{M}). I ... 0answers 62 views ### Checking whether a given matrix has a non-zero determinant For a positive integer n, let c be the number of ordered integers tripartitions (a_j,b_j,c_j) of n. Now consider the c \times c matrix M in which the value of the M[i,j] is M[i,j]={(... 0answers 38 views ### Is that possible to use stieltjes transform for multiple matrices I have the matrix calculation with expression $$\frac{1}{M}tr(\mathbf{WHH}^H\mathbf{W}^H + \mathbf{R}_{nn})^{-1}$$ whereas \mathbf{H} \in \mathbb{C}^{M\times K}, \mathbf{... 0answers 40 views ### Gramian of a permutation group orbit let W\in R^{d\times k},d>k. Suppose that the associated gramian has the following structure:$$ W^TW=(P_{1}t,\cdots,P_{k}t) $$with the set \{P_{i},i=1,\cdots,k\} forming a group of ... 0answers 89 views ### Does this set of (structured) equations always have a solution? Let r_1,\ldots,r_K be arbitrary positive numbers. Does$$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in \...
Let $U = diag([U_1, U_2, ..., U_N])$ be a block-diagonal $NM \times NM$ unitary matrix, where each $U_j$ is a unitary $M \times M$ matrix. Furthermore, let $Q_e = kron(I_M, Q)$ be the Kronecker ...