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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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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Equivalence of system of nonlinear equations

Let $A\in\mathbb{R}^{n\times n}$ be a semi-positive definite, $b\in\mathbb{R}^n$ and $k>0$. Consider the system of nonlinear equations $$ (1) \quad Ax=-k\frac{x}{g(x)}-b. $$ Let $A^+$ be the ...
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48 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
24 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
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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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19 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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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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2answers
35 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
23 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
122 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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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

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
10 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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3answers
43 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
9 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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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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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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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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1answer
27 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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23 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
14 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 ...
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1answer
22 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
14 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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1answer
10 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 $\sum\limits_j t_{ij} = 1$ $\forall$ ...
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1answer
27 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
48 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 $v$. ...
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1answer
20 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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1answer
39 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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0answers
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Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
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19 views

Find 3 eigenvectors and rife vectors of rank 1 matrix

I am trying to solve this without have to factor a polynomial of degree three of higher. $$ \begin{pmatrix} 2 & 4 & 2 \\ 4 & 2 & 4 \\ 2 & 4 & 2 ...
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33 views

Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior(alternative) tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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1answer
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solve problems using linear systems [on hold]

Leanne works at a greenhouse store. she needs to plant a total of 32 bulbs. two types of bulbs are available. she is asked to plant 3 times as many crocus bulbs as tulip bulbs. how many of each should ...
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2answers
31 views

Kernel of a bounded linear operator on a normed linear space need not be closed or open?

How should be the kernel of a bounded linear operator on a normed linear space as a set? Kernel of a bounded linear operator on a normed linear space need to be closed or open? Or it need not be ...
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0answers
31 views

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row?

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row(uniform distribution)? What sort of algorithm should I use to do this task? Brute Force algorithm- ...
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26 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
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4answers
259 views

If a matrix has positive, real eigenvalues, is it always symmetric?

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric? A class of symmetric ...
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2answers
29 views

Express $x+y+z$ in terms of $a$ and $b$ [on hold]

If $A = X + Y$ and $B = X + Z$, find the value of $X+Y+Z$ in terms of $A$ and $B$.
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1answer
59 views

Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
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1answer
108 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
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28 views

Find a matrix which maximizes expression

Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$ A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy $$ How can I find the matrix $A$ that maximizes expression ...
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1answer
30 views

Isomorphism between $E$ and $E^*$

Show that there does not exist a isomorphism $\phi:E\rightarrow E^*$ that it takes every basis of $E$ to its dual basis. ($E$ is a vector space over field K and $\text{dim}E=n$ .) My attempt: There ...
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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, ...
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1answer
32 views

Reformat this math formula to not need parenthesis

I've got a pricing equation that I am working with for an eCommerce site and I need to reformat this equation to not need parenthesis. Original Formula: {price} + ({length} * ({ppf} + ...
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1answer
32 views

Finding the standard basis for $\mathbb{R}^4$ that can be added to the set $\{(1,-4,2,-3),(-3,8,-4,6)\}$ to produce a basis for $\mathbb{R}^4$

Finding the standard basis for $\mathbb{R}^4$ that can be added to the set $\{(1,-4,2,-3),(-3,8,-4,6)\}$ to produce a basis for $\mathbb{R}^4$. I first check that the two vectors in the set are not ...
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1answer
51 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
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0answers
25 views

Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
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40 views

Solving a specific equation involving cos and sin

Here is the equation: $a\sin(\alpha+2\theta)+b\sin(\beta+\theta)=0$, where $\theta$ is the variable, $a$ and $b$ are positive, $\alpha$ and $\beta$ are constant. Please help and thank you very ...
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3answers
45 views

Finding the Characteristic Equation

For the following matrix I need to find $$\begin{bmatrix}-3 & 2 &1 \\3 & -4 & -3 \\-8 & 8 & 6 \end{bmatrix}$$ a. Characteristic Polynomial of $A$ b. Eigen Values c. Eigen ...
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31 views

Orthogonal vectors and potential

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthogonal vectors ? $c \in \mathbb{R}$ is a constant, and $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
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2answers
25 views

Prove $\exists$ $v \in V$ so that $(v , f(v))$ is a basis of $V$

maybe you guys can help me with this one. Let's say we have a vector space $V$ with $dim(V) = 2$ and we have a linear map $f : V \rightarrow V$ with $f^2 := f \circ f = 0$ ...