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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External operation: binary and unary perhaps???

Consider the following examples from which some definitions are derived: Let us take an element from the set R of real numbers (say, the number 8) and another from the set L of lengths (say, 4m). ...
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2answers
310 views

Derivative of Hadamard product

What is the derivative of Hadamard product of two matrices with respect to one of them? I.e. what is $D(AB)$ with respect to $A$?
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7 views

Radial Basis Function on 2 dimensional data

I have 2 dimensional point x=(x1,x2). I want to apply Radial Basis Function on this 2D data and transform it to the infinite dimensional space. could any one help me that what will be the new data ...
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29 views

Determining kernel and image of linear map

Problem: Which of the following maps are linear? Determine the kernel and the image of the linear maps and check the dimension theorem. Which maps are isomorphisms? 1) $L_1: \mathbb{R} \rightarrow ...
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15 views

What formula would I use for a four factor prioritization method where the factors are summed and ranked?

We are developing a way to prioritize system issues. Our current ranking is 1 - 5, but that becomes rather flat when dealing with a couple hundred issues. In our new method, we have four factors in ...
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22 views

Gradient and invariance under change of basis

I intuitively would be inclined to believe that the gradients $\nabla F_i$ of the components $F_1,\ldots,F_3$ of a vector field $\mathbf{F}:A\subset\mathbb{R}^3\to\mathbb{R}^3$, $\mathbf{F}\in ...
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10 views

Radial Basis Function on 2dimensional data

my data is a 2 dimensional point:x=(x1,x2). i want to apply RBF kernel on my data and transform it to infinite dimension space. i know what will be its infinite dimension form when data is defined in ...
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710 views

Determine point of interesction of plane with axis given points of plane

Q: The points $(2,-1,-2)$, $(1,3,12)$ and $(4,2,3)$ lie on a unique plane. Where does the plane cross the z-axis. I understand that the point of intersection would occur at $(0,0,z)$ and I have ...
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43 views

When does a matrix have short vectors in its kernel?

Consider an $n$ by $n$ matrix $M$ whose elements are in $\{0,1\}$, say. Now consider all vectors $v \in \mathbb{Z}^n$. Is there any mathematical property of $M$ which expresses when the kernel of ...
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1answer
36 views

An exercise question in Linear Algebra Done Right by Axler [duplicate]

Prove or give a counterexample: if $U_1$, $U_2$, $W$ are subspaces of $V$ such that $V$ = $U_1\oplus W$ and $V = U_2 \oplus W$, then $U_1 = U_2$. I'm a beginner in linear algebra and I'm ...
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35 views

Class of matrices for wich $A^T=J-A.$

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $$A = A^{\top}.$$ Instead, a matrix of ones or all-ones matrix is a matrix ...
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147 views

Solve system of equations for the ratios of the vectors

(Sorry for the bad title, didn't think of a better way to describe the problem). I have a system $\mathbf{A}\in\mathbb{C}$ that forms the problem $\mathbf{Ax}=\mathbf{b}$, for which I want to find an ...
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4answers
508 views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ ...
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1answer
34 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
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21 views

Expanding vector norm into sum

I'm trying to expand a simple Euclidian vector norm into a sum of $x_i$ coefficients, so that for each $i$ term, I can treat everything as coefficients for a quadratic. I think I must have messed up ...
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1answer
40 views

Curl: invariant under change of basis or not?

I wondered how the curl$$\text{rot}\mathbf{F}=\left( \begin{array}{ccc}\partial_y F_3-\partial_z F_2 \\ \partial_z F_1-\partial_x F_3 \\ \partial_x F_2-\partial_y F_1 \end{array} \right)$$of a vector ...
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2answers
27 views

Reduced row echelon form without introducing fractions at any intermediate stage

How can I reduce this matrix to reduced row echelon form but without using fractions in intermediary steps (I can use them in elementary row operations just not in the results in the matrix) $$ ...
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4answers
40 views

proof that every finite matrix has an annihilating polynomial

I don't quite understand the proof my notes gave me. Dimension of $n$ by $n$ matrix is $n^2$. Hence if $k \geq n^2$ then $\mathbf{ \{ I, A, A^2, ..., A^k \} }$ is linearly dependent. So, there exist ...
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45 views

Not sure if my Math Operation right or wrong. [on hold]

Hello guys I'm stuck with this. I'm not sure if my math operation right or wrong(Sorry for my bad English) Equation located here http://1.1m.yt/Cl3lGSbEn.png m=z=>>0?(z1-z2%50):(z1-z2%50)-50
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1answer
355 views

Defining an inner product from a norm which satisfies parallelogram law

Suppose we define inner product on complex inner-product space as the following : $$ \langle u,v\rangle =\frac{\|u+v\|^2 - \|u-v\|^2 + \|u+iv\|^2i - \|u-iv\|^2i}{4}$$ Given that the norm satisfies ...
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0answers
15 views

Put, in matrix form: $t_i=\sum_{j=1}^n \frac{w_j-w_i}{1+e^{-(x_i-x_j)}}$, $\forall i=1,2,…,n$

I have the set of equalities $$t_i=\sum_{j=1}^n \frac{w_j-w_i}{1+e^{-(x_i-x_j)}}, \ \ \forall i=1,2,...,n$$ and I try to write them in a more concise form. I tried to do so: $$t_i=\sum_{j=1}^n ...
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2answers
32 views

Proving existence and uniqueness of a matrix,

Let A be nxn with real coefficients and assume that it has n distinct eigenvalues, and all eigenvalues are positive real numbers. Let k $\ge$3 be an odd integer. a) Prove there exists a unique real ...
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1answer
54 views

Does it make sense to talk about complex matrices over the field of real numbers, R?

I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers. But I wanted to ask, ...
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1answer
60 views

Generalized eigen vectors of real symmetric matrix

From generalized characteristic equation, $Av=DBv$, where $A$ is a real symmetric matrix, $D$ is a diagonal matrix containing the eigenvalues as diagonal elements, and $v$ is the eigen vector ...
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2answers
71 views

Incomplete Circulant matrix

The eigenvectors and eigenvalues of a Circulant matrix are well-known to be related to the discrete Fourier transform of entries of one row (the exact terms are given here). My question: is there any ...
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1answer
377 views

Determine homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $y = 2x – 6$

Determine the homogeneous transformation matrix for reflection about the line $y = mx + b$, or specifically $ y = 2x – 6$. I do $mx - y +b =0$: $\text{slope} = m$, $\tan(O)= m$ ...
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1answer
26 views

Determinant proof question.

Using determinants, prove that if $A_1,A_2,...,A_m$ are invertible $nxn$ matrices, where $m$ is a positive integer, then $A_1A_2...A_m$ is an invertible matrix. Need help starting the proof. Do I ...
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2answers
42 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
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2answers
37 views

Determinant Question.

Show that if $A=\begin{bmatrix}a & b\\c & d\end{bmatrix}$, then $\det(A)=\frac{1}{2}\det\left(\begin{bmatrix}1 & 1\\tr(A^2) & (tr(A))^2\end{bmatrix}\right)$. I tried finding the ...
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1answer
21 views

The Maximum Eigenvalue of $F\mathrm{max(B)}F^T - FBF^T$

$F$ is a $b \times n$ real matrix. $B$ is a $n \times n$ real matrix, constructed by $B = w^T w$, where $w$ is a row vector with strictly positive real numbers, and clearly $B$ is a rank 1 matrix. ...
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2answers
25 views

Matrix multiplication and determinant question

Show that if $\det(\begin{bmatrix}b & c\\a & b\end{bmatrix})=0$ with $A=\begin{bmatrix}a & a\\b & b\end{bmatrix}$ and $B=\begin{bmatrix}b & b\\c & c\end{bmatrix}$ then ...
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1answer
14 views

how do I parametrise a stochastic matrix

I have a matrix $\mathbf{t}$ that maps one $d$ dimensional probability distribution to another $\mathbf{t}^T x = q$, i.e. with $\sum\limits_i t_{ij} x_i = q_j$ and $\sum\limits_j t_{ij} = 1$ $\forall$ ...
2
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1answer
42 views

Restriction of a linear algebra to an affine subspace?

Let's assume $V$ and $W$ are finite dimensional vector spaces and, $F:W\longrightarrow V$ is a one-to-one affine map i.e, $F(W)$ is an affine subspace of $V$. Also, let $T:V\longrightarrow V$ is a ...
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2answers
30 views

Block Matrix Zero Determinant Implication?

Recently I've been working with a number of square (order of 2n) matrices whose determinants are zero. That is, $$\det\begin{bmatrix}A&B\\C &D \end{bmatrix} = 0$$ where each of A,B,C, and D ...
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3answers
153 views

Linear dependence of these functions?

How can I check if these three functions (which belong to vector space $R^R$) are linearly dependent: $$e^{2x}, e^{3x}, x$$ If I take $\alpha, \beta, \gamma ∈ R$ and write the linear combination as: ...
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1answer
30 views

Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$-independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I ...
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0answers
9 views

How to build an 2-D polynomial from 1-D orthogonal polynomials

I have an set of orthogonal polynomials such as I want to build an 2D polynomial following the equation $$P_k(x,y)=P_k(x)P_k(y)$$ where $k=1..4, (x,y) \in [-1, 1]^2$ Based on given $P_n(x)$ as ...
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1answer
22 views

Variance of subset vs total variance

Is it true that the variance of subset is smaller than variance of the total set? Given each element in the set is a N-dimensional vector, and the distance is defined as Euclidean distance. Variance ...
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3answers
49 views

Let $v_{1}=(1,-2,3),v_{2}=(0,-1,2)$. Enlarge $\{v_{1},v_{2}\}$ to a basis for $\mathbb{R}^3$.

For instance, let $v_{1}=(1,-2,3),v_{2}=(0,-1,2)$. The set $\{v_{1},v_{2}\}$ is linearly independent. Enlarging $\{v_{1},v_{2}\}$ to a basis for $\mathbb{R}^3$ I simply form a matrix using ...
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1answer
26 views

Which polynomial has similar properties with Legendre?

I am looking for an kind of polynomial such as Legendre properties that polynomial sequence of orthogonal polynomials such as bellow image. Could you suggest to me one polynomial? Is B-spline correct? ...
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1answer
12 views

dimension of the quotient of a bialgebra

I am stuck in a proof of a lemma that I am in need of. The situation is as follows: Let $k$ be a field and $A$ and $B$ two finite-dimensional $k$-bialgebras, where the dimension of $A$ is a prime ...
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1answer
12 views

How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
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0answers
21 views

Rank of the sum of two rank 1 matrices, proof check

Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that $uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$. That is, for any vector $u$ of ...
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1answer
30 views

Solutions to the equation $a + b - ab/t = t/2$

$$a + b - ab/t = t/2$$ Where $0 < a < b < t$, $a,b,t \in \mathbb{N}$ and t is even, ie $t\mod2 = 0$ What are the possible values for a, b for a given t? For example, if t = 1000, then a = ...
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26 views

is this conclusion true or false?

Let $\mathcal{A}$ be a factor Von Neumann algebra and $\Phi$ is a map on $\mathcal{A}$ which is injective and surjective and $\Phi(0)=0$. If $A, B, C \in \mathcal{A}$ and ...
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1answer
24 views

Why is $[\alpha]_{\mathfrak{B}}=P[\alpha]_{\mathfrak{B'}}\rightarrow\alpha'_{j}=\sum_{i=1}^{n}P_{ij}\alpha_{i}$ obvious?

In the middle of looking into one of the theorems regarding coordinates a part of the proof of the one that I was reviewing at that time—which is presented below—puzzled me in that it was not so ...
3
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1answer
29 views

Find the inverse of a specific Vandermonde matrix

Let $$ V=\begin{bmatrix} 1& 1& 1& \cdots& 1 \\ 1& \xi& \xi^{2}& \cdots& \xi^{n-1} \\ 1& ...
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1answer
28 views

Proving that $V = U_1 \oplus U_2 \oplus \ldots \oplus U_k$.

Problem: Let $V$ be a vectorspace and $\beta$ a basis for $V$. Now make a partition of $\beta$ in a disjoint union of subsets $\beta_1, \ldots, \beta_k$ and let $U_i = \text{span}(\beta_i)$ for every ...
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0answers
40 views

Proving linear algebra equation [on hold]

I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent. (This problem comes from Bayesian estimation, ...
3
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0answers
53 views

Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$.

I want to prove the following statement: Let $A \in \Bbb R^{n\times n}$ with eigenvalues $\lambda$ and eigenvectors $v$. Show that $A^k$ has eigenvalues $\lambda^k$ and eigenvectors $v$. ...