All Questions
Tagged with matrices matrix-theory
248 questions
3
votes
1
answer
4k
views
Eigenvalues of product of symmetric positive definite matrices
Let $T_1, \ldots, T_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$.
Can I say
$$
\frac{x^T T_1 T_2 \ldots T_n x}{x^T x} \geq \mu^n
$$
If these matrices ...
- 41
3
votes
1
answer
207
views
Better name for “vec transposition permutation matrix”?
Let the operator vec($A$) unroll all the elements of $A$ into a single column vector in column-major order. Then, the elements of vec($A^T$) are a permutation of the elements of vec($A$). If I want to ...
- 208
3
votes
1
answer
3k
views
Solving a vector of quadratic equations
I have a system of $n \times 1$ equations
$$
0 = A\,vec(xx^t) + B x + C
$$
where
$x$ is a $n \times 1$ vector of unknowns
$x^t$ means transpose
$vec$ means $xx^t$ has been vectorized so has dimension ...
- 43
3
votes
2
answers
354
views
Solving linear matrix equation
Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
- 31
3
votes
1
answer
102
views
Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)
Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$.
For monics $...
- 10.5k
3
votes
1
answer
375
views
Closed-form expression for differential of matrix function
Let $X$ be a real $n\times n$ positive semidefinite matrix of rank $m\le n$ and let $Y\in\mathbb{R}^{m\times n}$ be the unique matrix satisfying (i) $X=Y^\top Y$, and (ii) $Y\, [I\, |\, 0]^\top = L$ ...
- 2,712
3
votes
1
answer
739
views
Operator norm of difference of matrix decompositions
This question is in part related to a question that I have already posed.
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
- 55
3
votes
1
answer
101
views
Isomorphism concerning $Soc(M_n(R))$
It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
- 355
3
votes
1
answer
187
views
Eigenvalues of certain matrices
We write $R(\theta)=\left(\begin{smallmatrix}\cos(2\pi\theta)&\sin(2\pi\theta)\\ -\sin(2\pi\theta)&\cos(2\pi\theta)\end{smallmatrix}\right)$ for any $\theta\in\mathbb R$.
Let $d,m,n,r$ be a ...
- 2,446
3
votes
1
answer
1k
views
Completing the square of a matrix expression
Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
- 2,712
3
votes
1
answer
240
views
Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$
Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. We find an upper triangular basis matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ of $\Lambda$. Is $B$ unique up to the right action of $\...
- 1,813
3
votes
1
answer
234
views
Is there a matrix that has the completely opposite effect of a Hadamard matrix?
First, let me provide some background on the problem:
In the field of Large Language Model quantization/compressions, outliers (abs of outliers are much larger than the mean of abs of all elements in ...
- 31
3
votes
1
answer
144
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
3
votes
1
answer
2k
views
Eigenvalues of a block matrix with zero diagonal blocks
Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix
\begin{equation}
M:=\begin{pmatrix}
0_{k_1} & A\\ A^\top & 0_{k_2}
\end{pmatrix},
\end{equation}
...
- 53
3
votes
2
answers
346
views
If $S$ is a nonsingular symmetric matrix over a number field and $D_k$ is its principal minor of order $k$, is $\frac{D_k}{D_{k-1}} > 0$ always true?
In Chapter II, Paragraph 4, Section 1 of
F. R. Gantmacher, The Theory of Matrices. Vol. 1 English translation by K. A. Hirsch. Chelsea Publishing Company. 1959. SBN 8284-0131-4,
the following ...
- 131
3
votes
1
answer
290
views
Bandwidth reduction of multiple matrices
Suppose I have a symmetric matrix $A$ and several diagonal matrices $D_1,D_2,\dots$. Are there any matrix transformations such as $P^\top A P$ so that
$$P^\top A P, P^\top D_1 P, P^\top D_2 P, \...
3
votes
0
answers
142
views
Solvability of a matrix exponential equation - generalized matrix logarithm
For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation
$$
G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .
$$
Basic ...
- 1,081
3
votes
0
answers
255
views
Homotopicity of $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ as morphisms from $A$ to $A\otimes A$
let $A$ be a $C^*$ algebra. We equip $A\otimes A$ with the spatial norm.
Assume that two morphisms $a\mapsto a\otimes 1$ and $a \mapsto 1\otimes a$ are homotopic morphisms, i.e, there is a curve $\...
- 356
3
votes
0
answers
359
views
Do we know what the impulse to "introduce" the Jordan canonical form was?
Mo-ers,
Do you know how it was that the study of the Jordan canonical form began?
There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
- 87
3
votes
0
answers
122
views
Algebra of block matrices with scalar diagonals
I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
- 3,041
3
votes
0
answers
65
views
How to show that a continuous family of symmetric matrices is uniformly positive?
My problem : I have a family of $4 \times 4$ symmetric matrices. More precisely consider an interger $d$, a real $\lambda> 0$ and define the family $S_{\lambda}$:
$ \{A(\lambda,x_1,x_2) ; (x_1,...
- 31
3
votes
0
answers
178
views
On a matrix inequality based on the Schur-Horn theorem
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.)
Let $A_s$ denote the symmetric part of $A$...
- 2,712
3
votes
0
answers
1k
views
Eigenvalues of block-hermitian matrices with zero diagonal blocks
I have a matrix of the form
$$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$
where $C$ is not necessarily hermitian. In general, can we say anything about the ...
- 47
3
votes
0
answers
56
views
Equivalence Classes of a Subgroup of Similarity Transformations
Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices
$$
\begin{bmatrix}
A & B\\
C & D\\
...
- 263
3
votes
0
answers
481
views
"Natural" ways of interpolating unitary matrices
Given two unitary matrices $A$ and $B$, that are "near" each other in some sense (perhaps $\left\lVert A-B\right\rVert <\epsilon$ for some norm, what are some sensible ways to interpolate between ...
- 757
3
votes
0
answers
70
views
Condition number after some "non standard" transform
Given a positive definite matrix $A$, and a diagonal matrix $B$ with positive diagonal entries, is the following inequality generally true?
$$\kappa((A + B)(I + B)^{-1}) \leq \kappa(A)$$
$I$ is an ...
3
votes
0
answers
611
views
Can anyone help me deduce a matrix inequality?
The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying $F^{...
- 103
3
votes
1
answer
740
views
Finding an adjacency matrix whose cube's diagonal is equal to a given vector
How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and
$$\mbox{diag}\left(A^3\right)=b$$
for some given vector $b$?
Is there a way to characterize all ...
- 503
2
votes
4
answers
293
views
Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle
...or prove that none exists.
Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory).
Preferably the ...
- 21
2
votes
2
answers
421
views
On matrix norms
It is standard to define an induced matrix norm $|||\cdot|||$ from a vector norm $||\cdot||$ in this way:
$|||A|||=\max_{x \neq 0}{\frac{||Ax||}{||x||}}$.
Suppose we define a different function of ...
- 6,962
2
votes
1
answer
316
views
Classification of congruent integer matrices
I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known ...
2
votes
2
answers
565
views
Regarding minimal elementary generators for $GL(n, \mathbb{Z})$
I have a result concerning the minimal number of elementary generators (and by this I mean generators which are elementary matrices) for $GL(3, \mathbb{Z})$. I'm currently working on extending the ...
2
votes
2
answers
307
views
The structure of the $n$-th power of a special matrix
Assume the following matrix
$$
C_p^{(a,b)}:=\left(
\begin{array}{cccccc}
a &a &0 &\cdots &\cdots &0 \\
0 &0 &a &\ddots &\ddots &\vdots \\
\vdots &\ddots &...
- 313
2
votes
1
answer
714
views
Is there a natural distance between skew hermitian matrices?
Working in machine learning, I try to find a way to compare time series, which can be considered as semi-continuous matrices belonging to $\mathbb R^{n \times \mathbb R}$ (a column corresponds to n-...
- 103
2
votes
1
answer
264
views
Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
- 637
2
votes
1
answer
137
views
Existence of matrices with some invertibility properties
Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$.
I am interested ...
user83947
2
votes
1
answer
184
views
Condition for non-vanishing trace
Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix.
Question: Does there always exist a symmetric $n \...
- 2,712
2
votes
2
answers
123
views
Behavior of matrix rank under thresholding of its elements
Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is ...
- 2,246
2
votes
2
answers
117
views
Powers of small square matrices over the Laurent polynomial ring with integer coefficients
I'm trying to calculate the powers of a 2 by 2 matrix with entries in $\mathbb{Z} \left[ t,t^{-1} \right]$.
The matrix is \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix}
I tought of writing my ...
- 205
2
votes
1
answer
344
views
Joint convexity of trace of matrices
Let $\Gamma_{m\times m}$ be a diagonal matrix with positive diagonal entries and $\mathbf{A}_{m\times m}$ be an arbitrary matrix. Then, is the following trace function jointly convex on $\Gamma_{m\...
- 287
2
votes
1
answer
157
views
Under row operations and column permutations a matrix A can be put in the non-unique form ( I | X ), what is known about the set of possible X?
Given a full row-rank matrix $A$, this can be put into a unique reduced row echelon form via elementary row operations. Allow column permutations (no column addition / multiplication) and this can be ...
- 33
2
votes
1
answer
156
views
Minimal Laplacian spread of a graph
Laplacian spread of a graph is the difference among the largest and the second smallest Laplacian eigenvalue of the graph. Is there any result or conjecture that discusses about the graphs having ...
2
votes
1
answer
236
views
An inequality regarding projection
Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b << 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define ...
- 482
2
votes
1
answer
456
views
Constrained optimization over a trace functional
Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$
$$\begin{...
- 2,712
2
votes
1
answer
303
views
Submatrix with small sum of elements
Let $A$ be an $n \times n$ matrix, for which I know the size of the sum of all its entries. Now I want to select an $m \times m$-submatrix, whose sum of entries is as small as possible. Is there any ...
- 2,197
2
votes
1
answer
2k
views
power of a block triangular matrix
I have a matrix in the form :
$$M =
\begin{pmatrix}
A & 0 & 0 \\\
B & A & 0 \\\
C & D & A
\end{pmatrix}
$$
where $A,B,C,D$ are diagonalizable square matrices and I want to ...
- 21
2
votes
0
answers
72
views
Gradient descent over the set of complex symmetric matrices
In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation:
$$ \...
- 21
2
votes
0
answers
146
views
What are the name and inverse of an interesting integer matrix?
It is practicable to compute the matrix inverses
\begin{align*}
\begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 2 & 2^2 \\
\end{pmatrix}^{-1}
&=\begin{pmatrix}
1 & 0 &...
- 1,091
2
votes
0
answers
137
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 823
2
votes
0
answers
502
views
Finding a basis for the range of a linear function
I realize this question is not high level but I have posted it on Math Stackexchange:
Stackexchange question
and have received some upvotes but no answers or comments, so I am trying here.
I will need ...
- 121