Tagged Questions
0
votes
1answer
33 views
A matrix has a real logarithm if it has a positive spectrum.
The title is a proposition I read in my notes that's left with no proof. Where can I read one?
0
votes
0answers
49 views
Definition of $\exp(A)$ in terms of spectral decomposition.
I am read this question Plugging a matrix multiplied by an imaginary number in the exponential function. Here the question'author defined $$\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$$
What ...
1
vote
3answers
109 views
A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?
Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve
$$\min_{X \in \mathbb R^{n ...
6
votes
2answers
154 views
An “Itzykson-Zuber”-like integral
I been told that there exists an integration formula, which states (or something of this sort)
$$
\int_{U(N)} dU \det[(\mathbb I+XUYU^{-1})^{-r}\propto ...
1
vote
1answer
59 views
Book about applied linear algebra to the 3D world?
I have no problem understanding the basics of the linear algebra, but I also know that the linear algebra it's still an analytical approach to real world scenarios that can be solved with matrices and ...
2
votes
1answer
80 views
Construction of Hadamard Matrices of Order $n!$
I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power.
Since
$$
n!-1 \bmod 4 = ...
1
vote
1answer
48 views
Good source for self study of matrix decompositions
What is a good source for study of various types of matrix decomposition, which is both comprehensive and also includes applications? It should at least cover LU, RQ, SVD, spectral, Schur, and ...
8
votes
1answer
102 views
The Determinant of a Sum of Matrices
Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$,
\begin{align}
\det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma},
\end{align}
...
0
votes
0answers
41 views
Ring of Row Finite matrices
I need to study rings of row finite matrices. But I can not find a book that exposes this theory. Can someone recommend me a good book to this theory?
2
votes
2answers
193 views
Require brilliant resources to self teach.
I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics: ...
3
votes
0answers
232 views
Good introductory book for matrix calculus
Hi I am an electronics graduate and working on image processing for the past one year...I have a basic exposure to linear algebra(thanks to Gilbert Strang..!!!). Now I am facing problems with matrix ...
0
votes
0answers
22 views
examples of or references for the computation of norm of circulant matrices
I search examples of circulant matrices such that the entries are roots of unity and such that the spectral norm is known.
Recall that the spectral norm of $A$ is the square root of the largest ...
4
votes
4answers
204 views
Book on matrix computation
I'm taking a machine learning course and it involves a lot of matrix computation like compute the derivatives of a matrix with respect to a vector term. In my linear algebra course these material is ...
3
votes
1answer
60 views
Skew-triangular (?) matrices and their properties
I'm asking this just out of curiosity because a brief googling failed to give me the answer.
By skew-triangular matrices I mean matrices with this
$$
\begin{bmatrix}
\times & \times & \times ...
5
votes
1answer
165 views
Polynomials in matrices with integer entries
I'm looking for references, if there is any, for this problem:
Characterize all elements $a \in M_n(\mathbb{Z})$ for which we have $\mathbb{Q}[a] \cap M_n(\mathbb{Z})=\mathbb{Z}[a].$
Here, by ...
1
vote
1answer
42 views
Need help to understand some definitions
I found following two definitions in a Ben Israel's book whose title is
Generalized inverses: Theory and applications:
For any $A, B \in \mathbb{C}^{m\times n}$, define
$R (A, B) = \{Y = A X B \in ...
1
vote
3answers
200 views
The matrix exponential: Any good books?
Looking for a book/article with a lucid exposition of the matrix exponential, preferably including the case of infinite matrices. Basic properties especially, but also differential equations are of ...
0
votes
0answers
38 views
Matrix separability preservation under conjugation!?
Someone know any paper about matrix separability preservation under conjugation? A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words:
$C(P_{1} ...
6
votes
2answers
222 views
Reference for a derivative formula for matrices
I found the following identity:
$$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$
on the matrix cookbook. It is equation 47 on page 8. Note that $X$ is an $n \times ...
4
votes
1answer
90 views
Parabolic subgroups of $\mathrm{Sl}_n$ are the ones that stabilize some flag
I am looking for a reference for the above statement that every parabolic subgroup of $\mathrm{Sl}_n(\Bbbk)$ stabilizes some flag in $\Bbbk^n$. I have gone through a large pile of books and can't seem ...
7
votes
0answers
119 views
Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$
What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$?
$\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
3
votes
3answers
136 views
the transformation which rotates a matrix by a half turn
Consider
$$T_{2}: \left[ \begin{array}{cc}
a & b \\
c & d \\
\end{array} \right] \rightarrow
\left[ \begin{array}{cc}
d & c \\
b & a \\
\end{array} \right]
$$
$$T_{3}: \left[ ...
4
votes
3answers
118 views
Is there a standard operation to “rotate rings on matrices”?
Is there some standard operation to "rotate rings on matrices"? Look at the image below:
The numbers around the four empty squares are what I'm calling ring, In the second matrix, this ring has ...
1
vote
1answer
49 views
Explicitly write down $g\in GL(n,\mathbb{C})$ so that $gAg^{-1}$ is upper triangular, where $A\in M_n(\mathbb{C})$
This is an elementary question which is do-able by hand but I am actually looking for suggestions or book references since I am sure that someone did this somewhere:
suppose
$$
A = \left( ...
1
vote
2answers
119 views
Resources for matrices and its applications
I was preparing some presentation slides on basics of matrices and its application. Even though, many of the participants are familiar with basic matrix operation, I planned to explain them by ...
1
vote
1answer
128 views
Index notation clarification
Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw ...
0
votes
0answers
102 views
Sylvester's law
Is there an elegant proof of Sylvester's law of inertia available online anywhere? It was set as an exercise in my textbook and I have proved it some time ago but now for the exams I want to know a ...
3
votes
1answer
154 views
Need help to understand a theorem
I have been reading a theorem related with the existence of the outer generalized inverse of a matrix where i have certain difficulties to understand the theorem.
Theorem is as follows.
Let ...
0
votes
1answer
62 views
About $P_{{L},{M}}$, projection transformation onto subspace $L$ along subspace $M$ .
I need help to study following theorem:
For every idempotent matrix $E\in\mathbb{C}^{n\times n}$, $R(E)$ and $N(E)$ are complementary subspaces
with $E = P_{{R(E)},{N(E)}}$. Conversely, if $L$ and ...
0
votes
0answers
56 views
paper about linear independence in altered Vandermonde and Cauchy Matrices
Both Vandermonde and Cauchy matrices with $n$ rows and $k$ ($n \geq k$) columns have the property that any $k$ rows are linearly independent (assuming the coefficient are independent). It seems to me ...
3
votes
2answers
162 views
Rewriting automorphism of matrix algebra in terms of automorphisms of the underlying ring?
I've used the following idea as a black box for some time now, but it occurred to me I don't fully understand why it's true.
Suppose $A=M_n(R)$ is the algebra of square matrices over some division ...
4
votes
6answers
600 views
A book for self-study of matrix decompositions
I am a third year math student and I noticed that there are many uses for decomposing a matrix (I mean decompositions like SVD, LU etc').
Is there a good book for self-study of the subject ?
Note ...
3
votes
2answers
372 views
Classification of simple modules for algebra of upper triangular matrices?
I've been refreshing my linear algebra, and this is a question of curiosity I have.
Let $U:=U_n(F)$ be the algebra of upper triangular $n\times n$ matrices over a field $F$. Is there a classification ...
4
votes
1answer
101 views
Does anyone remember a paper or talk: “Some matrices I have known”?
Quite a few years ago I seem to have read a paper or heard a good talk with the title
"Some matrices I have known". Does anyone recall that, or can give a reference?
3
votes
1answer
1k views
Computing the derivative of a quadratic form and matrix chain rule
I'm working on using the Generalized Method of Moments to analyze some yogurt purchase data, and in the course of trying to implement the standard Hansen method (i.e. not an empirical likelihood ...
4
votes
1answer
220 views
Holomorphic function of a matrix
A statement is made below. The questions are:
(a) Is the statement true?
(b) If it is, does it appear in the literature?
Here is the statement.
For any matrix $A$ in $M_n(\mathbb C)$, write ...
1
vote
4answers
156 views
Is there a classic Matrix Algebra reference?
I'm looking for a classic matrix algebra reference, either introductory or advanced.
In fact, I'm looking for ways to factorize elements of a matrix, and its appropriate determinant implications.
...
2
votes
1answer
66 views
Is the result of adding several positive semidefinite matrices also positive semidefinite?
I have a certain number of nxn matrices that are positive semidefinite. Is the result of adding all these matrices also positive semidefinite? If affirmative, is it always the case or, instead, in ...
1
vote
0answers
53 views
formulas for exact values of singular values in low dimension?
Are there formulas for the singular values of a real matrix in low dimension, i.e. for a $2 \times 2$ matrix or a $2 \times 3$ matrix?
Any comment is welcome.
2
votes
1answer
53 views
Eigenvalues of the matrix $(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$
$M_{[i],[j]}=(-1)^{i_1+i_2+\cdots+i_k+j_1+j_2+\cdots+j_k}$, where $1\le i_1<i_2<\cdots<i_k\le n$ and $1\le j_1<j_2<\cdots<j_k\le n$, can be taken to be an $\left(n\atop ...
3
votes
1answer
677 views
How to construct magic squares of even order
Could someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?
1
vote
0answers
71 views
Second eigenvalue of a stochastic block matrix
Considering a stochastic block matrix in the form of,
\begin{equation}
\textbf{$P_{}$} =
\left( {\begin{array}{cc}
\textbf{$A_{}$} & \textbf{$B_{}$}~; \
\textbf{$B_{}$} & \textbf{$A_{}$}
...
3
votes
1answer
328 views
Right Inverse of a matrix
I'm reading Linear Algebra by Bill Jacob and am having trouble with his development of the theory behind the right inverse of a matrix. I did an internet search but didn't find anything useful. Does ...
3
votes
2answers
537 views
The rank of skew-symmetric matrix is even
I know that the rank of a skew-symmetric matrix is even. I just need to find a published proof for it. Could anyone direct me to a source that could help me?
12
votes
4answers
384 views
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
I was reviewing some matrices and found this interesting
if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
0
votes
2answers
152 views
Ask for references on the comparison of $|A\circ B|$ and $|A|\circ| B|$
Let $A,B$ be complex matrices of the same size. I am looking for some references on the comparison of $|A\circ B|$ and $|A|\circ| B|$, where $|A|=(A^*A)^{1/2}$, "$\circ$" stands for Hadamard product.
...
2
votes
1answer
305 views
Most readable / accessible engineering math textbook
I'm looking for a book to learn engineering mathematics from. I graduated from college around 15 years ago, but I'm thinking about taking some sort of technical engineering masters degree. (I haven't ...
1
vote
0answers
104 views
Is that series-transformation known in the context of divergent summation
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
1
vote
2answers
459 views
Reference for matrix calculus
Could someone provide a good reference for learning matrix calculus? I've recently moved to a more engineering-oriented field where it's commonly used and don't have much experience with it.
14
votes
4answers
344 views
Powers of random matrices
Let $M$ be an $n \times n$ matrix whose elements are random reals in [0,1].
Two questions.
What is the growth rate of the magnitude of the elements of $M^k$ as a function of $k$? It is definitely
...
