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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Are two nilpotent $3\times3$ matrices with rank 2 always similar?

If we have any two nilpotent $3\times3$ matrices $A, B$ with rank $2$, are they always similar? I started by considering their nullity, which is $1$. Then, since they are nilpotent, their only ...
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30 views

“General position” in a poset

Does anyone have a reference for this notion of "general position" in a poset: A set $S$ in a poset $(X,\le)$ is said to be in general position if for any $A,B\subseteq S$, $\{x:\forall y\in B,y\...
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How do the eigenvalues of $(X^{T}X)$ help understand the amount of variability in prediction in regression?

In my courseware on an introduction into forecasting, we are given the formula for a $\alpha \%$ prediction interval being $$\mu \pm c_{\alpha}\hat{\sigma}\sqrt{1 + x_{0}(X^{T}X)^{-1}x_{0}}$$ for the ...
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24 views

Trying to learn about kernel PCA but cannot understand some math.

I'm trying to learn about kernel PCA by reading through the paper of it's creators (I assume) "Nonlinear Component Analysis as a Kernel Eigenvalue Problem", Bernhard Schölkopf, Alexander Smola, Klaus-...
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23 views

Projection matrix of $X$ multiplied by a column of $X$

Define the projection matrix of $X \in M_{n \times k}(\mathbb{R})$ by $$P_X = X(X^{T}X)^{-1}X^{T}\text{.}$$ Suppose $X$ is partitioned by $X = [X_1 \mid X_2 \mid \cdots \mid X_k]$ (so each $X_i$ is ...
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19 views

Obtaining constant using the straight line gradient of two equations

My question is the following: We have two equations, $$\left\langle R^2_j \right\rangle=\frac{1}{N_j}\sum\limits_1^{N_j}R^2_{i,j}$$ and $$ \Delta^*_j=\frac{kT}{2D\sigma}[1-exp(-2\frac{\Delta_j}{\...
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17 views

Proof regarding restriction of an endomorphsim

I need help with the proof of this Theorem: the restriction of an endomorphism $f:V\rightarrow V$ to a vector subspace $H \subset V$ such that $H$ is invariant is still an endomorphism: $f|_{H}: H\...
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20 views

Operators problem

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. Having $$L=\frac{1}{2}\left(\frac{\partial}{\...
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28 views

Polynomial basis - two possible ways to solve?

Is the following set is a span of $\mathrm{P}_{3}(x)$, $K = \{ x^3, x^2-x, x-1, 1\}$? Is it right to say that if we can express $\sum_{i=0}^3a_{i}x^{i}$ with $\sum_{i=1}^3b_{i}\mathrm{k}_{i}$, where ...
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25 views

Linear Functionals

I am stuck along the way while trying to prove that there exists a nonzero functional $f$ on $V$ such that $T^tf=cf$ for any scalar $c$ and a nonzero $\alpha\in V$ s.t. $T(\alpha)=c\alpha$. Where $V$ ...
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42 views

Determinant multiplication (det(ab)=det(a)det(b))

https://proofwiki.org/wiki/Determinant_of_Matrix_Product I found this proof (Proof 2).Could you tell me two things? 1: How do I know that if a matrix is invertible then it is a product of elementary ...
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17 views

The relationship of rank and multirank of a tensor

I have a tensor $T$ of size $2 \times 2 \times 2$ that has multi-ranks including: Column rank is $2$, row rank is $1$; mode-3 rank is $2$. What is rank of tensor $T$? On the other hand, what is the ...
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Prove that $\frac{1}{[Z^{-1}]_{kk}}=\frac{\text{det}Z} {\text{det}Z_{kk}}=\text{det}Z_{kk}^{\text{SC}}$, $Z_{kk}^{\text{SC}}$ is the Schur complement

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading ...
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28 views

Is Richardson iteration algorithm backward stable?

How do we analyze the numerical stability of Richardson iteration algorithm? Can we say it is backward stable?
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33 views

Can a matrix be orthogonal with respect to a positive definite $\phi$?

I've a doubt on othogonal matrices. I know that an orthogonal matrix is a matrix $O$ such that $O^{T}O=O^{-1}O=I$ and also that $O$ has on the columns and on the rows the coordinates of the vectors ...
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36 views

What does a eigenvalue equal to 0 represent?

The only relation I can think of is that the determinant of corresponding matrix A is 0 so A is not invertible. Could anyone suggest any other properties related to eigenspace, orthogonal ...
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13 views

$Ax=y$ with a inequality condition Prove every $x_i$ is positive

Let $A=(a_{ij})\in M_n(\mathbb R)$. Let $x =(x_1,\cdots,x_n)^t,\ y=(y_1,\cdots,y_n)^t$ be vectors satisfying $Ax=y$ and $$\sum_{j\ne i} \max\{a_{ij},0\}<y_i\le a_{ii}+ \sum_{j\ne i}\min\{a_{ij},0\}...
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14 views

Calculating inner product in integral form on an example

Given the inner product $(u|v)$ as: $(u|v) = \int_{0}^1 \overline{u(x)}v(x)e^x dx + \int_{0}^1 \overline{u'(x)}v'(x)\, \mathrm{d}x \qquad$ Where $v(x) = v$ So given that $u = u(x) = x$ i get for ...
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38 views

Solving an equation involving a matrix and a vector

I have to solve the following equation for x:$$ \sum_{k} (x'A_kx - 1)(A_kx)=0$$ $A_k$ is a matrix and $x$ is a vector. Can anyone give me some suggestion on how to solve this problem? Thanks in ...
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23 views

off-diagonal elements of projection matrix

I have a question on projection matrices. If $A$ is a projection matrix(i.e. $A$ is symmetric and $A^2=A$). I want to prove that $|a_{ij}|\leq1/2$ for $i\neq j$. I proved that the diagonal elements $...
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15 views

Linear Algebra and Quadratic Equations

I'm just wondering if Linear Algebra is concerned only with Linear equations? Can quadratic equations(or any higher power) also be considered under Linear Algebra? What does the term Linear stand for?
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29 views

Determinant of a jacobian

I have the following problem.The jacobian matrix is given in the image below.I just cannot seem to figure out how they arrived at the determinant.Can anyone show the steps or elaborate the procedure?
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35 views

Young tableaux to Specht polynomial to Irreducible representation for $(1,3,5) \in S_5$

What I am trying to do? Work out the irreducible representation of the group element $(1,3,5) \in S_5$ for the partition $2+2+1$ . Motivation: Learn how to calculate irreducible representation from ...
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47 views

Linear equations over finite field of size 2

Let $\alpha_1^1x_1+\ldots+\alpha_n^1x_n=1$ $\ldots$ $\alpha_1^mx_1+\ldots+\alpha_n^mx_n=1$ are equations in $\mathbb{F}_2^n$.(It is not system of equations). I am trying to prove that if every ...
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26 views

Convexity of the set of symmetric forms whose sum of k smallest eigenvalues is nonnegative

Let $V$ be an $n(<\infty)$ dimensional real vector space. Let \begin{array}{l} W:=\left\{\sigma\in V^\ast\otimes V^\ast\ |\ \sigma\text{ is a symmetric form on }V\right\}\oplus\left\{\tau\in V\...
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41 views

Finding solution to matrix equation over GF(2) with minimal true variables

I am looking for a general way to find a solution to a system of equations in GF(2) such that the solution has the least amount of true variables. After Gaussian elimination I get a matrix like such: ...
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48 views

Dimension of subspace of commuting matrices

I was hoping for some confirmation of a proof to the following preliminary exam question: Fix an $n \times n$ matrix $A$ with entries in an algebraically closed field $k$. Let $C$ be the space ...
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37 views

Why is the dimension of $k$-Tensors that are antisymmetric $\binom{n}{k}$

I'm trying to prove it but so far I can only write the antisymmetric tensor using the basis for all $k$-tensors. Can you please help me? Let $V$ be a vector space and $\{e_1, ...,e_n \}$ it's basis ...
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Kernel and cokernel of certain applications in $L^p$ spaces.

Let $X$,$Y$ two finite-dimensional closed linear subspaces of $L^p := (L^p)^n$ (defined in a finite measure space). Define $$L^p_X = \{ f \in L^p : \int \langle f,x \rangle = 0 \quad \forall x \in X \}...
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in a system of Linear equations why is the inequality in the numbers of equations to that of the unknowns give me a no or an infinite solution set?

From my understanding, in a given linear system represented by $Ax=b$, $A$ is the amount of each vectors, $x$ the vectors making up the system and $b$ the vector that I'm trying to reach through my ...
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28 views

Difference between Scalar Multiplication and Orthogonal Projection

My textbook is in Swedish so I am sorry if the wording is a bit weird or if things have got another name in english. Anyway, I am confused regarding Scalar Multiplication and Orthogonal projection, ...
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Can one find the signature of a real symmetric matrix just from the signs of some minors?

Background to the question: We know that a real symmetric matrix has all the eigenvalues $>0 ( \ge 0)$ if and only if all the diagonal minors are $>0 (\ge 0)$. Also, one can tell the number of ...
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32 views

If all the diagonal blocks are not of the same order then all the eigenvalues are not an integers.

Let $M=\begin{pmatrix} \begin{array}{cccccccc} 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\ 1 & 1 & 0 & 0 & ...
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34 views

Diagonalizable Operators and Matrices Characterizations

I'm trying trying to prove diagonalizable criterion for operators and matrices. It's a little lengthy: Let $V$ have basis $\{e_1,...,e_n\}$ and fix $T\in\scr{L}$$(V)$. I know for any matrix $A(T)$ of ...
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45 views

Recommendations on visualizing basic linear algebra

I am teaching linear algebra this semester, and I would really like to recommend my students some cool youtube videos visualizing some simple stuff like the span of a set of vectors, linear dependence,...
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Change of basis matrix with basis vectors multiplied by constants

I'm to write a change of basis matrix from basis e to f, where e is the standard basis and f is given by: $f_1 = 1/\sqrt{3} * \underline{e}(-1, 1)$ $f_2 = 1/\sqrt{2} * \underline{e}(1, 1)$ So the ...
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$T$ cyclic and $\mu_{T}(x) = p(x)^m $ where $p(x)$ irreducible $\Rightarrow $ T indecomposable

Definitions: $\cdot \hspace{0.4mm} \mu_{T,v}$: unique monic polynomial of least degree contained in $\text{ann}(T,v)=\{f(x) \in \mathbb{F}[x]: f(T)(v)\}$. $\cdot \hspace{0.4mm} \mu_{T}$: unique ...
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41 views

Difference/When to use Row and Column Vectors

I have a few questions about row vs. column vectors (MSc CS Thesis): What is the standard in computer science (if not mentioned explicitly)? Row or column vectors? If I write "vector $\mathbf{x}$ is ...
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Any matrix can be reduced to a “special” matrix by elementary operations

Definition. Let $A$ be an $r \times s$ integer valued matrix. $A$ is "special" if there exists an integer $k$ such that $a_{ij}=0$ unless $i=j$ and $i \leq k$ and $a_{ij} \neq 0$ if $i=j \leq k$. ...
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Is the basis for a row space the same as the basis for the column space of the same matrix but transposed?

I'm hoping someone could tell me the difference, if there is one, between the basis of a row space, and the basis of a transposed column space. Or, given a matrix A, is the basis for Row A the same ...
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Is this a typo, or something I don't understand?

Might be typos... I'm being asked to find kernel, range, and nullity; there's nothing wrong with that, I know how to do it. But I'm at my wits' end with the meaning of the ones I highlighted in ...
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51 views

Classify the surface $x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0$

I am working on a problem in which I must classify the surface described by the following equation $$x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0.$$ I have looked at this Stack Exchange discussion (on ...
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24 views

For regular stochastic matrices, what power suffices to make it positive?

Suppose $P$ is a $N\times N$ non-negative matrix whose rows each sum to 1 (i.e. $P$ is stochastic). Suppose further that we know $\exists m$ such that $P^m$ has all positive entries (i.e. $P$ is ...
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57 views

What is this matrix operation/symbol?

What is this matrix operation/symbol, the $\langle 0 \rangle$ in this expression? $$\Large {\left(\mathrm{s}_{a1}^{\;\;\mathrm{T}}\right)^{\langle 0 \rangle}}^{\mathrm{T}} $$ I've looked around on ...
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Strang , Introduction to Linear Algebra, Clarification of Breakdown of Elimination Chapter 2

I am sorry in advance if my question is very simple, I am just a beginner. In Strang's "Introduction to Linear Algebra" in Section 2.2 page # 46 , he is explaining "Breakdown of Elimination", and he ...
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17 views

Rate of Koetter-Kschischang Codes

I'm studying this article on coding theory. Here $W$ denotes a $N$-dimensional vector space over $\mathbb{F}_q$. A code $C$ then consists of code words $V\in C$, where $V\subseteq W$ is a subspace of $...
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28 views

Solving a linear equation $Y=mX+c$ with $X,Y \in \mathbb{Z}$ but $m,c \in \mathbb{J}$ (are irrational)

I'm having trouble solving the equation as in the title. I'm seeking to determine if there exists a solution, and if so, to find the solution (which I have proved is unique, i.e. there is at most one ...
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24 views

Orthogonal, Normal, and Self-Adjoint operators

Let be $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ the operator given by $T=3ref_{x_2=2x1}$, where $ref_{x_2=2x1}$ is the reflection on the line $x_2=2x_1$. Define the operator $U:{(\mathbb{R}^2)}^n \...
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21 views

Linear Algebra, Invariant Subspaces

State if $M$ is invariant subspace for each of the following linear operators from $L$ to itself: a)$L=\langle e_1,e_2,e_3,e_4 \rangle,\quad M=\langle e_1+e_2+e_3+e_4,e_1+e_2−e_3−e_4 \rangle$, $D(e_1)...
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20 views

How do I decompose the 3-D idenity matrix into lower dimensions?

I have been comfortable writing things like $\mathbb{I}_{x,y}\otimes\mathbb{I}_{z}=\mathbb{I}_{x,y,z}$ but now I realize I have no idea how the actual mechanics of it work. My first, naive attempt was ...