Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Proving a matrix statement

Let $C$ be an $m\times n$ matrix in echelon form. Let $\textbf{e}_{j}$ denote the $j$-th standard basis vector in $\mathbb{R}^{n}$, so that the $i$-th entry of $\textbf{e}_{j}$ is $\delta_{ij}$. (i) ...
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Reference request: Tensor Product of $m$ $SU(N)$ algebras

I'm working on quantum mechanics of linear $SU(N)$ chain of sites. Specifically, I would like to study a Tensor product of $m$ Lie algebras $\mathfrak{s}\mathfrak{u}(N)$ $$ V\in SU(N)\\ W = ...
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symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix share the same eigenvalue

I am trying to show that the symmetric normalized Graph Laplacian and symmetric normalized Adjacency matrix share the same eigenvalues ($\lambda_i$ and $1 - \lambda_i$ for i=1 to n. $\lambda$ is an ...
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Multiple Linear Regression sample problem:

I am currently studying linear regression, but I am not sure I understand everything correctly. I was trying to solve some of the exercises at the end of my book, and I picked a random one below. I am ...
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5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
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matrices for linear map with different basis

Let suppose we consider GF(4) as a vector space over GF(2), and we have Frobenius map on GF(4). According to the representing Frobenius map by matrix, since we can define different basis for GF(4) ...
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Collision of two moving lines (3D)

I have two lines / edges moving with linear velocity in timesteps. How do I determine whether the lines collide / intersect in the intervening period? My lines are (P1,Q1) and (P2,Q2). The endpoints ...
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38 views

how to transform a quadratic equation into a matrix form?

I have this type of equation: $$ - a^ {2} A - \eta ^{2} B - a \eta C - b^{2} A' - \eta' ^{2} B' - b \eta' C' - a \eta' D - b\eta E $$ The capital letters, $A, A', B, B', C, C', D, E$ are just the ...
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Upper Hessenberg Form

I am given a matrix. I would like to reduce it to its upper Hessenberg Form. We are discussing eigenvalue computations in Numerical Analysis and the textbook just gives the algorithm for it without an ...
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Derivative of quadratic form w.r.t. matrix (product)

I need to show that some quadratic from: 1' A C A 1 is increasing in matrix C , where 1 is a (Kx1) vector of ones, and A and C are both (KxK) positive definite. Can I reason like this: 1) ...
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Extreme points by intersection of extreme rays and hyperplane

I just met one question and have no idea about the proof, hope someone can give me some ideas on how to attack this question. Given a graph $G=(V,E)$ with $|E|=n$. Define a set $S$: ...
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Decomposition of triangular matrix as Toeplitz matrices

How can I decompose a triangular matrix into a product of Toeplitz matrices or circular/Henkel matrices?
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Basic Matrix Transformation

The information I have is for a matrix transformation from R^3 to R^3 (denoted by L()), L(a_1) = 3(a_1) and L(2(a_1))= (5,-3,6). Find L(3a_1-22a_1), L(-4a_1), L(0), L(4a_1). What I tried to do was ...
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reflected object

At point $A= \begin {pmatrix} -3 \\ -3 \\5 \end{pmatrix}$ an eye is located, with viewing direction $v= \begin {pmatrix} 1 \\ 1 \\-1 \end{pmatrix}$ . An Object has his centre at point $O=\begin ...
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determine in what grid rhombus is a point

i have a rhombus ( i.e. diamond) grid determined by these equations ...
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19 views

quadratic form of orthogonal and skew-symmetric matrix

I'm having an equation to solve but I'm stuck at a form without known how to make it further. I have $A\in{SO(3)}$, what I mean is A is orthogonal matrix. $C=skew(\Omega)$ with $\Omega \in R^3$ I ...
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Is the square of a hermitian operator hermitian?

If $\hat{O}$ is hermitian, what I see is that $$(\psi,\hat{O}\phi) = (\hat{O} \psi,\phi)$$ Therefore, $$(\psi,\hat{O}^2\phi)=(\hat{O}\psi,\hat{O}\phi)=(\hat{O}^2\psi,\phi)$$ Therefore the answer to my ...
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Estimate vector sum with fewer vectors

Let $M$ be a $n \times 72$ matrix. Also let the $n \times 1$ vector $V$ represent the sum of all columns of $M$. How can I find a reduced set ($<< 72$) of columns of $M$ that best represents ...
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Is this triangle construction possible?

I must construct this triangle: Consider the triangle $ABC$. Take $D$ in the line of $BC$ such that $C$ is the mid point of $BD$ and take $Y$ in the line $AC$ such that the lines $AB$ and $BY$ are ...
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10 views

Values converging to the same Point

I have two points on a graph (4,3). Is it possible to get the two values to increment one value at a a time with a formula? Example: ...
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32 views

approximating a dominant eigenvector using power method

Question: Use the power method with scaling to approximate a dominant eigenvector of the matrix A. Use the Rayleigh quotient to compute the dominant eigenvalue of A. $A= \begin{pmatrix} ...
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Dual basis to $e_{i+1}-e_{i} \in \ker ((1,1,…1)^\vee\in(\Bbb E^{n+1})^\vee)$

Studying the root system $A_n$ given by the simple roots $v_i:=e_{i+1}-e_i \in \Bbb E^{n+1}/\Bbb R(1,1,...,1)$ for $i = 1,...,n$, I came across the following dual basis: $v_i^\vee:= ...
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Basis transformation matrix

Let $f: \mathbb{R^2}\rightarrow \mathbb{R^2}$ be a linear map. Let $$B = \left\{\left(\begin{array}{c}-1\\ 1 \end{array}\right),\left(\begin{array}{c}1\\ 1 \end{array}\right)\right\}, B' = ...
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For which values is the quadratic a perfect square?

$f(x)=ax^2+bx+c$ is a quadratic polynomial, $a,b,c$ are natural numbers. For which values of $a,b,c$, $f(x)$ can be a perfect square? If it can be perfect square, for which integer values of $x$ it is ...
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Fixed vectors in tensor product

Let $V$ be a finite dimensional $\mathbb{F}_p$-vector space and $f \in GL(V)$ and $x \otimes v \in \overline{\mathbb{F}_q} \otimes_{\mathbb{F}_q} V$. Let $\mathbb{Z}$ act on $\overline{\mathbb{F}_q} ...
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23 views

An endomorphism sending a basis element to zero

Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$. Let $u$ be the endomorphism $$u(P)=(X^2-1)P''-2XP'$$ I want to determine the determinant of $u$. So I proceed by ...
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18 views

reduced Hessenberg form

Suppose $A \in \mathbb{R}^{n\times n}$ is symmetric and has $k$ eigenvalues with $\lambda_1$ repeated $n_1$ times, $\lambda_2$ repeated $n_2$ times $\ldots$ $\lambda_k$ repeated $n_k$ times. So $n_1 ...
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36 views

Minimizing a vector constrained to a set

Sorry if this is wordy or over-complicated, I'm not sure how to isolate the problem any more than I have below without losing important context: I'm trying to implement a coordinate block descent ...
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33 views

Some questions about svd proofs and linear algebra

Theorem: The rank of A is r, the number of nonzero singular values. Proof: The rank of a diagonal matrix is equal to the number of its nonzero entries, and in the decomposition $A=U\Sigma{}V^*$, U ...
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Is condition number always a good measure for LS/ML problems?

Given is a system of linear equations $$ \mathbf{y} = \mathbf{A} \mathbf{c} + \mathbf{v} $$ where $\mathbf{y}$ is the N dim. observation vector, $\mathbf{A}$ an $N\times K$ matrix, $\mathbf{c}$ ...
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Connecting inner products and the trace of a matrix

Hello, I'm currently trying to work through this question. However, I am struggling with understanding what I should do. I am aware of all of the definitions used throughout, however I cannot link ...
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19 views

Dual Lefschetz Operator and Contraction with the Fundamental Form

Let $M$ be a Kahler manifold, with metric $g$, Kahler form $\omega$, Lefschetz operator $L$, and dual Lefschetz operator $\Lambda$. $\Lambda$ and contraction with $\omega$ both map $k$-forms to ...
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A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$ for all x

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
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29 views

Determine if the following linear transformation is surjective or injective

Let $S \left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = $ $\begin{pmatrix} x_1 & -2x_2 & x_3 & x_4\\ 2x_1 & - 4x_2 & -3x_3 & ...
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How can I compute pseudo determinant

Let A square n by n matrix and let b:=pseudo det of A And assume that A is diagonalizable and rkA=r Then what is pseudo det of AA^(t)??
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ANT Frohlich Proposition 3, (v). Induced map of dual modules has the same determinant

$R$ is a Dedekind domain, $V$ is an $n$ dimensional vector space over its quotient field $K$, $B(-,-)$ is a $K$-bilinear form on $V$, and $M, N \subseteq V$ are free $R$-modules of rank $n$. Also ...
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Linearization of multiple normal functions

I have noticed that it takes a very long time to perform non-linear least squares fitting on datasets similar to this: where there are multiple Gaussian distributions to be fit to experimental ...
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38 views

Simplifying a rather long expression

I'm struggling to simplify this expression: $$ i_1 j_0 \exp \left(\frac{-81 a^2-2 p_2 \left(65 a+49 b+57 p_2-114 x\right)+2 p_1 \left(16 a-65 b-49 p_2+49 x\right)+32 a b+130 a x-65 b^2+98 b x-65 ...
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Generalization of Vandermonde matrix for arbitrary powers

Suppose we fix arbitrary distinct powers $\alpha_i \in {\mathbb R}$, $i = 1,2,\ldots,n$. Under what conditions on the powers do any $n$ distinct choices of $x_j \neq 0$ give a basis $(f(x_j))_{j=1}^n$ ...
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17 views

Exact sequence of linear spaces

While reading Nigel Higson's book Analytic K-homology i found the result (which was known to me earlier, but I never saw the proof) that the index of the product of two Fredholms operators is the sum ...
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Decide coordinates for a vector in a triangle (Image attached)

I have the following triangle. I have to express the line $\overline{AT}$ as a linear combination of $\overline{AC}$ & $\overline{AB}$. A hint was to use the knowledge of $\overline{AT} = ...
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How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
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Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
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42 views

QR Algorithm with Shifts Question

Why must QR Algorithm with Shifts make no progress when applied to this n x n matrix? (attached as image). Also, if a matrix A is orthogonal in a QR factorization, will R be tridiagonal? How would ...
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Find Determinant of linear transformation

The question is Find the determinant of linear transformation Let V be the vector space of polynomials of degree at most over R, and define T:V to V by T(p(x))=p(1+x)-p'(1-x) for all p(x) in V. I ...
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29 views

Proof about dual bases?

Let V be a finite dimensional vector space over a field F. Let B={v1,v2, ..., vn} be a basis and consider the dual basis B*={v1*,v2*,...,vn*}. Let a be an element of V*. prove that $$v = ...
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Proving that $\mathrm{rank}(P_1+P_2) = \mathrm{rank}(P_1)+\mathrm{rank}(P_2)$

Supposing $P_1$ and $P_2$ two projectors as: $P_1\circ P_2 = P_2\circ P_1$. What is the condition for $P_1+P_2$ to be a projection? If it was the case above then how can I prove that ...
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18 views

Insights in solving systems of eqn?

So, I need to find all solutions in integers of the following system: $x_1 + x_2 + 4x_3 +2x_4 =5 $ $-3x_1 - x_2 + 0 - 6x_4 =3$ $-x_1 - x_2 + 2x_3 - 2x_4 =1$ I know the steps, but I don't ...
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When does the leading right eigenvector gives the stationary distribution?

I am trying to make sense of the meaning of the leading right eigenvector in mathematica modeling (applied mathematics). I am interesting in models of the kind $\overrightarrow v(n+1) = M ...
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24 views

mean number of links in adjacency matrix

I have converted from an individual-level adjacency matrix to one for clusters and I am trying to show mathematically how I programmed up determining the mean number of inter-cluster links. I am not ...