Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Atlas on the Grassmannian Variety

Let $G(k,n)$ the set of all $k$-dimensional sub-spaces of a vector complex space $V$ of dimension $n$. I know that it is possible to define the grassmannian as the quotient of $\chi(n,k)$ by $GL(k)$ ...
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68 views

Existing complete function space allowing discontinuity .

This is a question which came to me due to several previous question: sorry for the all previous links necessary to look to get the question. The latest question is in the link: Convergence on Norm ...
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60 views

Advantage/disadvantage of complete/incomplete metric space.

It must be simple. I understand a metric space can be complete for a given metric and and the same set may be incomplete with a different metric. This may be due to the fact that under the given ...
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52 views

Is it possible to compare Sobolev space and Polish space?

Is it very easy to say that Sobolev space and Polish space are unrelated? Or we can infer some connection or relation or common structure or generalize one to another? Any comment would be highly ...
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37 views

Simplification of squared dot product

Let $a=[a_1,\ldots,a_m]^T$ and $b=[b_1,\ldots,b_m]^T$ be two $m$-dimensional vectors in $\mathbb{R}^m$. I have this summation: $$\sum_{i=1}^m~a_i^2b_i^2$$ My goal is to find a function $f$, such ...
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Why are the eigenvalues of $I_R + \beta' \alpha$ bounded by one?

Trying to understand the Granger-Johansen Representation Thm (see p. 7 here; we are assuming Condition 5). We have $(p \times r)$ matrices $\alpha, \beta$. We know that $|eig(a \beta')| \leq 1$ and ...
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25 views

Deriving the Particular Solution to a Linear Discrete Dynamical System

In my lecture notes it says that for a linear dynamical system of the form $ f(x) = Ax $ where A is diagonalisable d x d matrix, with $ \left \{ v_1 , v_2, \cdots , v_d \right \} $ a basis for $ ...
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66 views

Conjugation in the complexification of a vector space switches its type

Let $V$ be a real vector space with an almost complex structure $J$ and consider its complexification $V^\mathbb{C}$ where we extend $\mathbb{C}$-linearly the linear maos of $V$, in particular $J$. In ...
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16 views

Linear Algebra quadratic forms (max and plot)

If I have $q(x)=x_1^2-x_1x_2-x_1x_3+x_2x_3$ How do I find the maximum value of $q(x)$ subject to the constraint $||x||=4$? I already know the max when $||x||=1$ since it is the eigenvalue, but I don't ...
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46 views

Multiplicity of an eigenvalue=number of times it appears on the diagonal?

Suppose $T \in L(V)$ and $\lambda\in F$. Prove that for every basis of $V$ with respect to which T has an upper-triangular matrix, the number of times that $\lambda$ appears on the diagonal of the ...
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138 views

Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
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Linear Transformation Similarity - Answer verifications

Let $B = \{1,x,x^2\}$ and $B^\prime = \{1, 1+ x, 1+ x + x^2\}$ and $T(p(x)) = p(x) - x\frac{dp}{dx}$. Find $[T]_B$ Find the transition matrix from $B^\prime$ to $B$. Find the transition ...
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How to graph quadratic forms and label points closest to and furthest from the origin?

$x_1^2+4x_2^2+9x_3^2=1$ $x_1^2+4x_2^2-9x_3^2=1$ $-x_1^2-4x_2^2+9x_3^2=1$ I have to sketch these three surfaces and determine which are "bounded", which are "connected", and what the points ...
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49 views

dimension of the span of all partial derivatives of a given polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...
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29 views

Graph Decomposition And Linear Algebra

A module in a graph $G$ is a subset $M$ of the vertices such that all the vertices in $M$ have the same neighbourhoods outside of $M$. That is, if $v_1, v_2 \in M$ and $x \not\in M$, then we have ...
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127 views

Coppersmith-Winograd algorithm

I'm interested in algorithms to compute matrix multiplications. Is the Coppersmith-Winograd algorithm similar to the Strassen algorithm ? I have two other questions: 1) Are the multiplications done ...
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65 views

Converting second order system into first order system (ODE)

The second order equation $\frac{d^2\vec{x}}{dt^2} = A\vec{x}\ + \vec{g}(t)$ models an earthquake's effect on a 7-story building. Let $x_j(t)$ be the displacement of the $j$th floor with respect to ...
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Solving a linear system in function of a parameter

Problem: Solve the following system in function of the parameter $b$: \begin{align*} \begin{cases} -bx + 2y - (2+b^2)z + bu &= -2 \\ x -2y + bz -u &= 0 \\ x + (2b-4)y + (2-b)z + (b-1)u &= ...
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Linear functionals defines projection operators?

The way I always understood linear functionals on a vector space $V$ is to consider then as measuring objects which give projections when they are given vectors. Now I wanted to make this a little bit ...
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104 views

How to solve an overconstrained system of equations?

What is the easiest codeable way to solve an overconstrained static model? How does Force Effect https://forceeffect.autodesk.com do it? Given a 10m long bar angled as the hypotenuse of a 3, 4, 5 ...
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Determining whether $\{f : \deg f = 0 \text{ or } \deg f = n\}$ is a subspace of the vector space of polynomials over a field.

Is the set $$W = \{ f(x) \in P(F) : f(x) = 0 \text{ or } \deg f = n \}$$ a subspace of $P(F)$ if $n\geq 1 $? The solution says it is not, as the set is not closed under addition. But if we ...
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Dual of Riesz basis with opt. stab. const. $\lambda_\min$, $\lambda_\max$ has opt. stab. const. $\frac1{\lambda_\max}$ and $\frac1{\lambda_\min}$.

Consider a Hilbert space. Consider a Riesz basis $\phi_k$, $k \in \mathcal{K}$ of this space, where $\mathcal{K}$ is an appropriate set of indices. By definition, the Riesz basis fulfils the ...
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25 views

Can we assume power spectrum of a time domain signal form a polish space?

Consider a time domain signal and we perform Fourier Transform on and get the power spectrum. Can we consider that the power spectrum comes from a polish space? It is a set from polish space?
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19 views

How to show these covariance functions form a kernel? (I.e. a covariance matrix for any finite set of points)

In machine learning (specifically, Gaussian processes), a "kernel" is a two argument function such that for any set of $N$ "input points," (any $N$, any points), the $N \times N$ matrix of pairwise ...
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59 views

Gauss Jordan Elimination different answers

I have a question regarding Gauss Jordan Elimination. I have this matrix: \begin{bmatrix}2&1&5&0\\1&0&-3&1\\7&2&2&1\end{bmatrix} So at the start I will switch R1 ...
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23 views

Degree of freedom in a system of inequalities

I have a system of inequalities of the form $Ax<=b$ where the unknowns $(x_1, x_2,..., x_m)$ outnumber the inequalities $(n)$. I know that in the worst case scenario $k$ inequalities become ...
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62 views

H-infinity methods in control theory and hardy space.

Sorry this is very simple but I do not know. Why the H-infinity methods that are used in control theory are said to work on Hardy space? If the question is not appropriate then how H-infinity methods ...
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Stabilizers of Segre varieties

What, if anything, is known about maps in PGL(V) that preserve Segre varieties? I am specifically interested in linear maps preserving the Segre embeddings of $\mathbb{P}^{15} \times ...
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Understanding the Frobenius Norm for Sparse Coding

I have a question regarding sparse coding, Non-negative sparse coding. Iterate until convergence: $ \mathbf{A_i} \leftarrow \arg \! \min_{A \geq 0} || \mathbf{X}_i - \mathbf{B}_i\mathbf{A}||_F^2 + ...
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52 views

Permutation as a product of generators of the permutation group

Let $G$ be a permutation group, generated by $g_1,\ldots,g_n$. And let $h$ be in $G$. Example: $G=\langle (12)(34),(123)\rangle$ and $h=(12)(34)(123)=(243)$ (reading the cycles from right to left, ...
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137 views

Eigenvalues and eigenvectors for earthquake modeling

My instructor explicitly stated that, because we are asked to find eigenvalues and eigenvectors of a $7\times 7$ matrix, MATLAB would be easiest to use. The equation $(1)$ is intended to resemble ...
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52 views

Non-unique factorization in $\mathbb{Z}[\sqrt{-5}]$

I want to show that the decomposition into irreducible factors in the ring $$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$ is not unique, except for the order of factors ...
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22 views

Parameterizing a linear compressor

I am hoping to build a function $f_{A,B,\alpha}(x \in \mathbf{R} ) \rightarrow y \in \mathbf{R}$ that serves as a positive signal compressor. The function acts on an input signal $x\left(t\right)$ one ...
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41 views

Can the following simple tensor preserving map exist?

In this question here, I asked if there could exist a $U \in U(4)$ such that $U$ itself was not the tensor product of two matrices, but such that $U(A \otimes B)U^{-1} = A' \otimes B'$ for all $A,B ...
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Express one constant as a linear combination of two other constants?

Let a,b,n,m,o be nonzero rational scalars. How do I express o as a linear combination of n and m with coefficients a and b? Explicitly how to find a,b such that an + bm = o? I know how to do linear ...
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65 views

Path-components of the general linear group using only elementary algebra

Let $E(c)$ be an elementary matrix of the type to add $c$ times a row to another row when applied to another matrix on the left (with $c$ in some off-diagonal position $(i, j)$), and, with the usual ...
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84 views

Rotate basis to align with vector

I have a coordinate system with the Basis $B=(e_x, e_y, e_z)$ and two vectors $r$ and $a$. Now, I want to rotate the basis so that the $e_x$ unit vector points in the direction of the vector ...
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60 views

Squaring a matrix using a linear memory

I have a N x N matrix (let's denote it with A). I want to calculate $A ^ 2$, using $\theta(N)$ memory (speed does not matter as long as it's a polynomial) on one processor. I believe that this can ...
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28 views

Expanding linear functional to base of $V^*$

Given a linear functional $f_1\in V^*$ where $V^*$ is a dual space of $V$, I can expand it to the base of $V^*$ : $B^*=\{f_1,f_2,...,f_n\}$, that I know. But does it mean that exist a base $B$ for $V$ ...
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linear transformation of orthogonal vector space on subspace

Let $V$ be a finite dimensional inner product space over $F$. If $W$ is a subspace of $V$, prove that the orthogonal projection of $V$ on $W$ is an idempotent linear transformation of $V$ into $W$. I ...
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135 views

Does this linear system of 5 unknowns and 2 equations have multiple solutions?

\begin{cases} x+ 2y - z + w - t = 0 \\ x - y + z + 3w - 2t = 0 \end{cases} Add 1st to the 2nd: $$2x + y + 4w - 3t = 0 \\ y = -2x - 4w + 3t = 0$$ Substitute y in the 1st: $$x - 4x - 8w + 6t - z ...
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Inverse properties of $L_1$ normed matrices

Let's take the adjacence matrix $A$ of a directed graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. (i.e. we divide each elements of the row by the sum of the elements of ...
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explicitly splitting the hamilton quaternions over local fields

For simplicity, lets first consider the hamilton quaternions $$ H = \left(\frac{-1,-1}{\mathbb{Q}}\right)$$ This is the central division algebra over $\mathbb{Q}$ with $\mathbb{Q}$-basis given by ...
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26 views

Unknown functions yield a given determinant

I am trying to develop a nomogram which simultaneously shows the exact Fisher equation $(1+u) = (1+v)(1+w)$ and its linear approximation $u \approx v + w$. This amounts to finding twelve smooth ...
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18 views

Minimal column sum (?) solution to system of linear equations over $\mathbb{Z}_2$

There are $n$ equations in $n$ unknowns where both the coefficients and unknowns come from the field $\mathbb{Z}_2$. I can represent these as the equation $Ax = b$ where $A$ is an $n\times n$ matrix ...
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50 views

SVD: How to find the columnvector of U corresponding to a singular value equal to zero

The question is if you have a situation where one of the singular values is equal to 0 in a singular value decomposition of a matrix, how to do you procede to find the column vector of U corresponding ...
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22 views

Representation of Affine Maps

I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$ f[\lambda x+ (1-\lambda) y]=\lambda ...
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48 views

How to compute the best fitting frustum for a set of points?

I am struggling with a problem that I am sure is well known, but I could not find any answer using google or searching on MathOverflow. I have a set of 3D points (x,y,z) and a camera reference frame ...
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41 views

What is meant by an eigenvalue of 2 matrices?

In looking for a way to compare covariance matrices, I came across a paper that formulates a metric using what appears to be a joint eigenvalue. I'm not familiar with this idea. Thus we propose ...
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40 views

How many distinct possible forms for its Jordan canonical matrix are there? 4x4 non-diagonalizable matrix with two unique eigenvalues

I know the sum of $A_m$ equals $4$ as $\dim(A) = 4$ and sum of $G_m$ can't equal $4$ as $A$ is non-diagonalizable. After I write down all the cases, what should I do?