For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
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1answer
29 views

How many subsets of unknowns whose sum can be determined by the underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$

Consider a underdetermined system $Ax=b$ with $A \in \{0,1\}^{m \times n}$ (i.e. being a binary matrix), $x \in \mathbb{R}^n$ and $b \in \mathbb{R}^m$. I want find a set $S$, $e \in S$ if and only if ...
0
votes
1answer
47 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
0
votes
1answer
23 views

Differentation of vector with respect the another vector [on hold]

$y$ is $m \times 1$ vector $y=Ax$. $A$ is $m \times n$ matrix in function of $z$. $x$ is $n \times 1$ vector in function of $z$. And $z$ is vector $r \times 1$. How can i find $\frac{dy}{dz}$? I ...
1
vote
1answer
20 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal

Let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to ...
3
votes
3answers
33 views

Intuitive understanding of vector / matrix calculcation

I am currently dealing with calculations done on vectors and matrices. For some parts I have gained an intuitive understanding, for others I didn't. E.g., when we are adding two vectors, you can ...
0
votes
2answers
1k views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
5
votes
2answers
1k views

$A$ is skew hermitian, prove $(I-A)^{-1} (I+A)$ is unitary

Given $A$ is a skew-hermitian, (i.e $A^H=−A$), the Cayley transform of $A$ is defined as: $W=(I-A)^{-1} (I+A)$. How can be proved that $W$ is unitary (i.e. $W^H W = W W^H = I$)?
5
votes
4answers
135 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
2
votes
1answer
31 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
4
votes
1answer
60 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ ...
0
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0answers
5 views

Difference between the SCM converging to the Marcenko-Pastur distribution and Johnstone's result about the top eigenvalue

I have a confusion which I suppose must be rather basic. As I understand, in the 60s/70s it was known that the empirical eigenvalue distribution of the sample covariance (of $n$ i.i.d. standard ...
0
votes
0answers
76 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix?
0
votes
1answer
41 views

Prove that a real matrix is a matroid

Problem $A$ real matrix, size $m\times n$ $M$ some structure, possible matroid $E(M)$ set of all columns of $A$ (we're considering them vectors) $I(M)$ set of all linearly independent columns of $A$ ...
1
vote
1answer
1k views

Does a Symmetric Matrix with main diagonal zero is classified into a separate type of its own? And does it have a particular name?

For example, I have a Matrix as shown below. Does this Matrix belong to a particular type. I am CS student and not familiar with types of Matrices. I am researching to know the particular Matrix type ...
1
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1answer
35 views

The root system of $sl(3,\mathbb C)$

I want to determine the root-system of the lie algebra $sl(3,\mathbb C)$. Does someone know a good (and complete) reference for this problem? I know that the root-system is $A_2$ but I want to see a ...
5
votes
0answers
30 views

Factor the matrix (scalar $\times A$) into permutations of $A$

Here's an example of $A . B = scalar \times C$, done with magic squares. The last square does not have a consecutive range of digits. Drop the magic square requirement. In $2\times2$ matrices we ...
0
votes
1answer
22 views

Operation count, LU-decomposition

I'm having trouble with an assignment question. The question is as follows: Determine the total number of multiplications and divisions (as a function of $n$) required to compute the LU-decomposition ...
10
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0answers
177 views
+200

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
1
vote
0answers
15 views

Find (linear) transformation matrix using the fact that the diagonals of a parallelogram bisect each other.

This is the first time I post something on this website. I'm on this question already for hours. I'm clearly not asking the community to do my homework, I'm hoping someone can explain me how I should ...
1
vote
1answer
20 views

L2 Matrix Norm Upper Bound in terms of Bounds of its Column

I need to find an upper bound for a matrix norm in terms of bounds of its columns. I have a vector $\varepsilon_i(x) \in R^{n\times1} $ such that $||\varepsilon_i(x)||_2\le\gamma_0$. I also have a ...
1
vote
1answer
34 views

Proving matrix is invertible using the Banach Lemma

I have an assignment question that goes like this: Consider the $n \times n$ matrix $$ \begin{pmatrix} 2 & 1 & 2^{-1} & 2^{-2} & 2^{-3} & 2^{-4} & \cdots & ...
0
votes
2answers
20 views

Finding a matrix inverse when an equation involving it is a multiple of the identity matrix

Say you had a matrix $A$, and you did an equation like $A^2 - A$, and proved that it was a multiple of $I$. How could you find $A^{-1}$ in the form $rA + sI$ after proving that? I want to do it ...
1
vote
1answer
13 views

Is it true that for all matrices $A$ and all traceless matrices $T$, there exists a traceless matrix $T'$ such that $AT = T'A$?

Fix a real number $n$. By a "matrix", I mean an $n \times n$ real matrix. Now let $A$ denote a matrix. Is it true that for all traceless matrices $T$, there exists a traceless matrix $T'$ such that ...
1
vote
0answers
32 views

Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
-1
votes
0answers
41 views

XOR binary matrix multiplication $AX=B$? [on hold]

Let $A$, $B$, and $X$ be binary matrices (in F2 ), where $A$ and $B$ are of size $n \times m$ with $n > m $. $X$ is an $m \times m$ matrix. Compute $X$ such that $AX=B$. ps: $A$ is not a ...
0
votes
0answers
11 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
0
votes
1answer
15 views

Prove of identity: $(Av) × (Aw) = CofA (v × w)$ [on hold]

How can I prove that for each $A \in M^{3×3}$ and $v, w ∈ \mathbb R^3$ $(Av) × (Aw) = CofA (v × w)$
0
votes
1answer
35 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{max}(B^{-1}A)}{\lambda_{min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
0
votes
2answers
13 views

Prove a covariance matrix is positive semidefinite

Given a random vector c with zero mean, the covariance matrix $\Sigma = E[cc^T]$. The following steps were given to prove that it is positive semidefinite. $u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = ...
3
votes
1answer
15 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
1
vote
0answers
50 views

Realation between Matrices ?!

The answer to the following question could be trivial. Let $A_1, A_2$ be symmetric $n\times n$ matrices, $x=(x_1,\ldots,x_n)\in \mathbb{R}^n$. If the maximum is taken for over ($\|x\|=1,\, and ...
0
votes
1answer
24 views

How polynomials are represented in matrix form for Univariate Polynomial. [on hold]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
0
votes
1answer
19 views

Prove that a small shift in the diagonal term leads to smaller spectral radius (for Perron-Frobenius theorem)

On Wikipedia, the proof for Perron Frobenius theorem in the strictly positive case has a confusing step: Suppose $T=A^m-\epsilon I$, where $\epsilon$ is smaller than the smallest diagonal term of ...
2
votes
2answers
172 views

Invertible skew-symmetric matrix

I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the ...
2
votes
1answer
2k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
0
votes
2answers
76 views

Row Reduce Augmented Matrix

I am having issues actually row reducing it. What I initially get for the augmented matrix is: \begin{pmatrix}\begin{array}{cccc|c} 0 & 1 & -2 & 1 & 2\\ 2 & -2 & 4 ...
0
votes
1answer
55 views

Wolfram|Alpha refuses to find the inverse of a large 6x6 matrix.

Just to be clear, this isn't a question on how to find the inverse of a matrix, I just don't want to find the inverse by hand (I hope you see why). $$ \begin{pmatrix} 1 & 2006 ...
-1
votes
0answers
16 views

show i-th projection is a linear transformation

For $i ∈ {1,2,...,m}$, define $\pi : F_m → F$ by $\pi(x_1,x_2,...,x_m) = x_i$ (the $i$-th projection). (a) Show that it is a linear transformation. (b) If $T : F_m → F$ is a linear transformation ...
0
votes
0answers
6 views

Why does “up to scale” make homograph matrix lose one freedom?

Can anyone explain "if H is up to scale, then dof(H)=8" in the following discussion? degree of freedom of Homography matrix Thank you!!!
1
vote
0answers
24 views

Derivative of an Euclidean-Vector norm.

Consider: x a $N \times 1$ vector , with elements $x_i$ b a $N \times 1$ vector , with elements $b_i$ A a $M \times N$ matrix , with elements $a_{ij}$ ( Symmetric matrix - Block Circulant ) As we ...
1
vote
1answer
35 views

Sylow's theorem for group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$

Let $G=SL(2,\mathbb{F_3})$ - group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$. (a) Find the order of $G$. I think it is $24$ but not sure how to verify it. (b) ...
1
vote
2answers
74 views

What is the enclosed volume of an irregular cube given the x,y,z coordinates of the 8 corners?

I have the xyz coordinates of 8 points that forms an irregular-shaped cube. This is an animation, so the cube is undergoing periodic or cyclical shape-change. The co-planarism of any group or set ...
0
votes
1answer
36 views

Linear Algebra. Is this question realte to combination and factorials?

I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks.
0
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0answers
20 views

Real Symmetric Positive Definite Matrix [on hold]

Assume $H = A + Bi$ is a positive $m \times m$ Hermitian matrix, where $A, B \in R^{m \times m}$. How can we show that $C = \begin{bmatrix} A & -B \\ B & A \end{bmatrix}$ is a real ...
0
votes
0answers
31 views

How to solve system of equilibrium probability state equations

I have started studying markov chains where i have these statistical equilibrium probability state equations.These equations are solved for a particular case $s_1=4,a_1=5,s_2=2, a_2=1$ and a 15*15 ...
0
votes
0answers
14 views

Is $X_{k+1}=\frac{1}{N}\sum_{i=1}^N \Pi_{X_{k}^{1/2}v_i}$ globally convergent?

Let $X_0=X_0^\top\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix, let $v_i\in\mathbb{R}^n$, $i=1,\dots, n$, be a set of $n$-dimensional real vectors and pick an integer $N>0$. I ...
0
votes
1answer
31 views

Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
0
votes
1answer
33 views

Multiplying magic squares like matrices to hopefully arrive at another magic square

Well, the title actually describes what is the problem in question. I was just thinking a bit about magic squares and this question popped-up. It could be that it is not interesting but I do not see a ...
0
votes
1answer
34 views

Linear algebra: Solving a system of equation matrix with a variable as coefficient.

Let's consider this augmented matrix $$\left(\begin{array}{ccc|c} 3 &-6 &6 &15\\ -2 &7 &a &-25\\ 2 &-6 &6 & 20 \end{array}\right)$$ I'm trying to ...
1
vote
1answer
46 views

Attempt to solve a matrix (counterbalancing) problem computationally gives “spooky” result: why?

This question is posted on the mathematics section of stackexchange because my uneducated guess is that the answer involves some basic mathematical principles, possibly in the domain of linear ...