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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1answer
10 views

$0 * \Lambda ^{1/2} \neq 0$ (?) Problems with matrix multiplication

Suppose $\Lambda$ is a diagonal matrix of size $n > 1$ and rank $1$, let's denote the sole element on the diagonal as $\lambda$. Consider the following equation: $\Lambda ^ {-1/2} = 1/\lambda * ...
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0answers
4 views

Stationary points with matrix

I have an exercise but I do not even know where I should start: Consider the normalised quadratic form $(x^T Ax)/(x^T x)$ where $x∈R^2$, $A$ is a general 2x2 matrix. Find the vectors that make this ...
4
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0answers
20 views

Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
0
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0answers
7 views

Find a basis for the column space - why not reduce to RREF first?

Related to Understanding how to find a basis for the row space/column space of some matrix A. . When asked to find the basis for the column space of a matrix, can I first reduce to RREF, and then use ...
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2answers
38 views

Find $\det(A)$ of Matrix and condition on a and b

Let $$ A=\begin{bmatrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{bmatrix} $$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
0
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1answer
15 views

L2 Norm of Inverse of Non-square Matrix Multiplication

Consider a matrix $A\in\mathbb R^{n\times m}$ with $n<m$. Given that $\|A\|_2 = \gamma_0$ and $AA^T$ is invertible, can we find any equality/upper bound for $\|(AA^T)^{-1}\|_2$ in terms of ...
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3answers
25 views

$A$ has more columns than rows and has full row rank, show there exist infinitely many $B$ s.t. $AB=I$

If A $\in M_{m\times n}(R)$ such that $n>m$. Prove that if $\text{rank} (A) = m$ then there are infinitely many matrices $B \in \ M_{n\times m} (R)$ such that $ AB = I_m$ So the question is ...
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0answers
49 views

Prove that $A+2I$ is invertible [duplicate]

Given $A$ is a square matrix such that $A^{3} = 2I$ Prove that $A+2I$ is invertible and find its inverse. How do I prove that $A+2I$ is invertible? For proving $A-I$ is invertible, I use ...
0
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1answer
25 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 ...
3
votes
3answers
35 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 ...
2
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1answer
32 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 $A$?
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1answer
21 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 ...
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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 ...
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0answers
78 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix?
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1answer
23 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 ...
2
votes
0answers
25 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 ...
2
votes
1answer
46 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 ...
0
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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 ...
4
votes
1answer
61 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
votes
1answer
26 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 ...
2
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1answer
18 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 ...
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vote
1answer
41 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 & ...
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0answers
33 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 ...
0
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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 $v, w ∈ \mathbb R^3$ $(Av) × (Aw) = CofA (v × w)$
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votes
4answers
171 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 ...
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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
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1answer
42 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 $A$ ...
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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] = ...
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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
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 ...
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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
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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 ...
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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 ...
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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!!!
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0answers
25 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
36 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) ...
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0answers
32 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 ...
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0answers
21 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 ...
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vote
2answers
47 views

Ring of matrices has no nontrivial ideals [duplicate]

It is a theorem that a commutative ring is a field if and only if it has no nontrivial ideals. Clearly this does not hold in the noncommutative case. I am trying to show for instance that the ring of ...
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0answers
18 views

Distance Geometry Problem (DGP) Programming Language Recommendation

We have been studying DGPs in clinic recently and I was hoping I might be able to get recommendations for computing languages in the processing of large network solutions. Specific computations ...
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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1answer
18 views

Determinant of block matrix with off-diagonal blocks conjugate of each other.

I am working on finding the determinant of the following block matrix $$ \begin{pmatrix} C & D \\ D^* & C \\ \end{pmatrix}, $$ where $C$ and $D$ are $4 \times 4$ matrices with complex entries ...
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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 ...
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0answers
35 views

Let $\operatorname{rank} A=1$ then there are, $x,y\in \mathbb{C}^n$ such that $A=xy^T$ [on hold]

Let $A\in M_n$ and $\operatorname{rank} A=1$. Are there $x,y\in \mathbb{C}^n$ such that $A=xy^T$?
2
votes
2answers
35 views

Positive definite matrix meaning in human language? “Definite”?

I have to consult Wikipedia every time to re-learn what is positive (semi) definite. So that I am sure I will be able to decompose it further in some ways. Wikipedia Now I am trying to truly ...
5
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0answers
26 views

Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear ...
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votes
1answer
45 views

How would I find the equation of a graph using matrices? [on hold]

Assuming the graph below is a 5th degree polynomial, how would I go about finding its equation using matrices? Edit: So if I had the data points: $A(2006, 531.37), B(2013, 484.13), C(2028, ...
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0answers
27 views

Eigenvalues of the subtraction of a gram matrix and a psd rank $1$ matrix.

If $V$ is $p \times r$ matrix ($r<p$) with rank $r$, then $G=V^TV$ is a $r \times r$ gram matrix with rank $r$. Let $x$ be a $r \times 1$ vector, and $xx^T$ is a psd rank $1$ matrix. It's already ...
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 ...