All Questions
Tagged with matrices linear-algebra
1,683 questions
6
votes
2
answers
236
views
Bounding the non-multiplicativity of isometric projection
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique Polar decomposition:
$A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$.
In particular the orthogonal factor is given by $$O_A=A(\...
- 6,741
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
6
votes
1
answer
293
views
When can I find a continuously-varying basis of eigenvectors for a non-simple eigenvalue?
This is related to my last question that went sadly unanswered (Is my matrix perturbation analysis legitimate?).
Again, I have a matrix whose entries are integer polynomials in a single real variable....
- 12.7k
8
votes
2
answers
950
views
Best known bounds on (border) ranks of small matrix multiplication tensors?
The $(m,n,p)$-matrix multiplication tensor is a representation of the bilinear map $T\colon\mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\rightarrow\mathbb{R}^{m\times p}$ given by $T(A,B)=AB$. ...
- 7,633
4
votes
2
answers
197
views
Requesting reference for result from linear algebra on Schur complements of M-matrices
In my research in linear algebra, I have come across a useful result stating that the Schur complement of a principal non-singular submatrix of an M-matrix is also an M-matrix, but I have never found ...
- 620
4
votes
1
answer
545
views
No arbitrary product of matrices has eigenvalue 1?
Consider the matrix $D$, adjacency matrix of an undirected graph $G$ on $n$ vertices, with the notation that $d_{i,i}=0,\forall i$.
The matrices $A_i$ are constructed from Identity matrices, $I_{n*n}$...
- 143
5
votes
1
answer
214
views
When does isometric projection respect multiplication?
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique orthogonal polar factor $O_A=A(\sqrt{A^TA})^{-1}$,
( $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$see Polar decomposition).
...
- 6,741
0
votes
1
answer
460
views
A question on orthogonal matrix [closed]
Let $P\in R^{n\times n}$ be an orthogonal matrix. I want to ask whether or not there exists some vector $x\in R^n$ containing no zero entries such that $Px$ also contains no zero entries.
- 377
1
vote
0
answers
154
views
When can a binary matrix be transformed into a certain form
I have a $k \times n$ matrix $G$ over ${\mathbb F_2}$ that's full rank.
This can always be put in systematic form : $G \sim [I_k \mid P]$ where $I_k$ is a $k \times k$ identity matrix and $P$ is a $k \...
- 451
4
votes
1
answer
758
views
Norm of Moore-Penrose inverse of a product
I had asked this question in math.stackexchange (link: https://math.stackexchange.com/questions/1902276/bounds-on-the-moore-penrose-inverse-of-a-product ) but I did not get any response so I am trying ...
- 549
3
votes
2
answers
2k
views
Matrix equation with Hadamard product and its own inverse involved
I know there is an almost exactly same question here but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\...
- 133
0
votes
1
answer
572
views
Recurrence Equation and Matrix Convergence
To begin with, let us give the conceptual background needed to expose the problem. First of all, we shall consider the set $\mathbb{L}^{n} = \mathbb{R}^{n}_{\geq0} = \{\overrightarrow{x}\in\mathbb{R}^{...
user89024
0
votes
1
answer
720
views
Unique solution to a matrix equations [closed]
Given any $n \times k$ real matrix $M$, where $n<k$ and $rank(M)=n$, I consider the following equation (where $M'$ is the transpose of $M$):
$$
MM' = MAM'
$$
Then clearly, $A = \mathbb{1}_k $, ...
1
vote
1
answer
76
views
On ranks of finite element matrices
Let $T=\{\alpha_1,\dots,\alpha_t\}$ be a set of distinct integers.
If $A\in T^{n\times m}$ is an $n\times m$ matrix with entries from $T$ then
does $rank_{\Bbb R_+}A=rank_{\Bbb Q_+}A$ hold true?
- 13.9k
1
vote
0
answers
55
views
On the Lowener-Heinz inequality
I know that for two symmetric positive semi-definite (non-diagonal) matrices $A,B$, the inequality asserts that the following does not hold for all $p > 1$
$$A \succeq B \succeq 0 \Rightarrow A^p \...
- 11
2
votes
1
answer
1k
views
When a Unitary matrix times its complex conjugate is diagonal
Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ (everything is taking place over $\mathbb{C}$). This is equivalent to a couple of other ...
- 928
0
votes
1
answer
432
views
standard form of antisymmetric matrix
Suppose $J$ is standard symplectic matrix =$\begin{bmatrix} 0 & I_d\\
-I_d & 0\end{bmatrix}$. Now $\theta=(\theta_{i,j})$ be any invertible $d\times d$ skew symmetric matrix. Then there ...
- 21
0
votes
0
answers
698
views
Singular Values of Linearly Combined Matrices
I have a question related with singular values of matrix sums.
Let's assume I have matrices $A$, $B$, and $D$ (positive, semi-definite) where $D = A + B$. For singular values of $D$, I know that
$$
...
- 101
6
votes
1
answer
205
views
Can a projective solvable group be transitive?
Let $p > 3$ be a prime number, and let $G \leq \mathrm{PGL}_2(\mathbb{F}_p)$ be a solvable subgroup.
Is it possible that the action of $G$ on $\mathbb{P}^1(\mathbb{F}_p)$ is transitive?
- 11.3k
0
votes
1
answer
170
views
Non-strict column diagonally dominant matrix inner product
Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:
$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...
- 323
2
votes
1
answer
303
views
Minimize matrix distance to tensor product
Minimize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product.
...
- 2,099
2
votes
1
answer
376
views
Maximize inner product of a tensor of unitary matrices
How can one maximize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$.
Both the maximum value of ...
- 2,099
1
vote
1
answer
201
views
nonnegative solution of nonhomogeneous under-determined linear system of equations
For a set of under-determined linear equations, I was wondering if there is any closed form for all non-negative solutions? Is there a way to analytically characterize the feasibility set of such ...
- 13
1
vote
1
answer
462
views
Is this a full rank matrix? [closed]
According to the answer of znt to the previous version, I revise the question as follows:
Is there a real $(n-1)\times n$ matrix $A$
such that $A$ is not a full rank matrix and satisfy $a_{ii}&...
- 356
1
vote
0
answers
167
views
Expected amount of linearly dependent random vectors? [closed]
Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$?
EDIT: As said in the comments, I'm looking for the ...
- 123
3
votes
0
answers
915
views
How to find a closed form of following matrix's determinant [closed]
I wanna find a closed form of determinant of the following matrix
$$A(n) =
\begin{pmatrix}
B_{1} & B_{2} & \cdots & B_{n} & 1 \\
B_{n} & B_{1} & \cdots & B_{n-1} &...
- 513
5
votes
1
answer
607
views
information measure for matrix that is analogous to rank
Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
- 163
2
votes
0
answers
53
views
Selecting columns from multiple matrices to form a well-conditioned matrix
Given multiple matrices of the same size, is there a way to select one column from each matrix to form a well-conditioned matrix?
For example, given four 4-by-10 matrices A, B, C, D (real, positive, ...
- 21
0
votes
1
answer
100
views
How does the rank of $C_i$ change with $i$?
Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ ...
- 133
0
votes
1
answer
2k
views
SVD alternatives for symmetric matrices
Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices
$$
A = E'E
$$
Practically this can be done easily using SVD ...
12
votes
2
answers
9k
views
What is the time complexity of the matrix exponential?
While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...
- 241
1
vote
0
answers
174
views
Negative eigenvalue of Toeplitz Hermitian matrix?
I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
- 495
5
votes
2
answers
389
views
Pfaffian of several skew-linear transformations / matrices
Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
- 53
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
6
votes
2
answers
825
views
Explicit solution to a Rayleigh quotient equation
For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
Main equation:
$$
\biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\...
- 163
5
votes
1
answer
780
views
Perturbations on the pseudoinverse of a matrix
Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation
$$
A_p = A + \Delta
$$
is there a way to represent
$$
(A_p)^{\star}= (A)^{\star} + f(\Delta)
$$
where $(A_p)^{\star}$ ($(A)^{\star}...
2
votes
1
answer
968
views
Eigenvectors of symmetric positive semidefinite matrices as measurable functions
I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices.
I've been searching everywhere for an ...
8
votes
1
answer
2k
views
Finding Toeplitz matrix nearest to a given matrix
For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz ...
- 495
6
votes
1
answer
996
views
Is there a smooth polar decomposition for non-invertible matrices?
Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite.
$P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$,
and when $A$ ...
- 6,741
4
votes
0
answers
374
views
non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?
We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e., the matrix is not only weak diagonal-dominant, but ...
- 329
2
votes
1
answer
360
views
Does similarity imply symmetric similarity?
Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that $A=QBQ^{...
- 367
-2
votes
1
answer
433
views
Eigenvalues of cyclic tridiagonal matrix [closed]
The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that this matrix ...
- 640
2
votes
1
answer
380
views
Bounding entries of the inverse of certain zero-one matrices
It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question:
Bounding the absolute sum of entries of the ...
- 1,081
1
vote
1
answer
2k
views
range of singular values of sub-matrices
Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and ...
- 351
2
votes
0
answers
783
views
Can the matrix exponential be equal to the elementwise exponential [closed]
Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential $e^A=\sum_{k\...
- 2,286
1
vote
3
answers
195
views
The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra
Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X)
\otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...
- 356
2
votes
1
answer
484
views
Modified interlacing of eigenvalues
Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A &...
- 135
3
votes
0
answers
531
views
How do I ensure that my matrix is positive definite? [closed]
I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form
\begin{equation}
M=\begin{pmatrix}
\Sigma&P'\\
P&\Sigma
\end{pmatrix}
\end{equation}
where $\Sigma$ is a $n\...
- 39
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
1
vote
1
answer
607
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 ...
- 1