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 ...

learn more… | top users | synonyms

0
votes
1answer
78 views

Matrix with only one invariant factor [duplicate]

Suppose $A$ is an $n \times n$ matrix over the field $F$ with only one invariant factor. Show any $n \times n$ matrix $B$ over $F$ that commutes with $A$ is a polynomial in $A$.
0
votes
1answer
38 views

Why does an augmented matrix with bottom-right 1 represent a system without solutions?

To determine whether a system of linear equations lacks a solution, we can convert it into reduced row echelon form. If the bottom-right value is 1, then the system lacks solutions. Why is this true? ...
0
votes
2answers
100 views

Exponential matrix question

Suppose the characteristic polynomial of $B$ is $$\displaystyle (\lambda_1 - x)^{b_1} \cdots (\lambda_h - x)^{b_h}$$ Using Jordan theory, show $e^B$ (the exponential matrix's) characteristic pol. is ...
0
votes
0answers
45 views

Deriving the fundamental equation relationship

I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic ...
-1
votes
1answer
363 views

spanning list and a linearly dependent set

Please help me understand this two notions: What is the difference between a spanning list and a linearly dependent set? and is there any relationship between the two?
1
vote
3answers
476 views

Finding trace and determinant of linear operator

I've got the following question Consider the linear operator of left multiplication by an $m \times m$ matrix $A$ on the vector space of all $m \times m$ matrices. Determine the trace and ...
1
vote
0answers
159 views

volume of linear transformations of Jordan domain

Let $T:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a linear transformation and $R\in \mathbb{R}^n$ be a rectangle. Prove: (1) Let $e_1,\ldots,e_n$ be the standard basis vectors of $\mathbb{R}^n$ (i.e. ...
2
votes
1answer
34 views

Show that the subpartitions $A_{12}$ and $A_{21}$ of a unitary matrix $A$ have identical singular values.

If a unitary matrix $A$ is partitioned as: $A=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$, where $A_{11} \in \mathbb{C}^{m \times m}$, $A_{22} \in \mathbb{C}^{n \times ...
0
votes
0answers
74 views

Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear ...
0
votes
2answers
135 views

Difference between Spanning set and Postitive Spanning Set

I do understand the difference as mentioned in the texts about spanning set and positive spanning set, im somehow missing how if $v_1.. v_r$ is a positive spanning set for $R^n$, then $v_2 ... v_r$ ...
1
vote
1answer
196 views

Similarity of Matrices

I might be wrong, but it seems to me that there are two notions of "simmilarity" with regards to matrices which are slightly different: A is similar to B if an invertible Matrix P exists s.t. $A = ...
1
vote
1answer
181 views

Determining the axis of rotation of a special orthogonal matrix

My syllabus states the following procedure to determine quickly the rotational axis of a 3x3 matrix $A$ that is orthogonal with determinant 1, but it's not completely clear: "If $A$ $\in$ SO(3) and ...
0
votes
2answers
578 views

What are actually proportional rows(columns) in a determinant?

So, I posted the question about determinant equality and thoght that two columns are proportional. But they are not. Please explain me this on the next exaple (I thought the first and the second ...
2
votes
0answers
179 views

Properties of Non-Diagonally Dominant Matrix

I have a question about properties of matrices which are or are not diagonally dominant. So I understand that a diagonally dominant Hermitian matrix with non negative diagonal entries is positive ...
6
votes
2answers
217 views

Vector Components - Superposition of Forces

I understand that by inspection of the given figure, $F_{1,x}$ (the x-component of $F_1$) must $< 0$, so $F_{1,x} = 250\color{red}{\cos{53^{\circ} }}$ can't be right. But I don't see how $F_{1,x} ...
0
votes
3answers
187 views

Find a $\lambda$ so the system has a unique solution?

$$\begin{align} 3x + \lambda y & = 5 + \lambda \\ 2x + 5y & = 8 \end{align} $$ I got that $\lambda$ can be anything by using Cramer's rule, so there are infinite solutions.
8
votes
3answers
327 views

Do complex eigenvalues of a real matrix imply a rotation-dilation?

This is part of a bigger proof that if there is a compact set, $K \subset \mathbb R^n$ such that the linear transformation $L$ maps $K$ into its interior, the eigenvalues $\lambda_i$ are all of ...
0
votes
2answers
153 views

Determine the minimum distance between lines

If anyone got time. Me and my friends appreciate the help We got a problem with the following task: Determine the minimum distance between lines Line 1: $$\begin{eqnarray*} x &=&1+t \\ y ...
2
votes
2answers
108 views

How to show for a PSD matrix $A$ that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$?

If $A \in \mathbb{C}^{n \times n}$ is positive semidefinite, show that $\left \| \left ( A+I \right )^{-1} \right \| \leq \frac{1}{1+\sigma _{\min}\left ( A \right )}$, where $\sigma _{\min}\left ( A ...
1
vote
1answer
2k views

Find a set of vectors $\{u,v\}$ in $\Bbb{R}^4$ that spans the solution set

Question: Find a set of vectors $\{u,v\}$ in $\mathbb{R}^4$ that spans the solution set of the equations: $$\begin{align}x - y - z + w = 0 \\ x + 2y - z + 3w = 0\end{align}$$ Reducing these I get: ...
2
votes
2answers
83 views

represent the matrix into rank 2

Given an $n\times 1$ vector $x$ and an $n\times 1$ vector $y$. The $n\times n$ matrix $xy^T$ is a rank one matrix. Now let $M=xy^T+yx^T$, how do we represent the matrix $M$ as a rank 2 form $M=AB^T$, ...
0
votes
3answers
75 views

Determining if a Number N can be written as sum of one or more numbers from 1,2…K each used at most once?

How to determine if a natural number N can be written as sum of numbers from 1,2...K such that each number 1,2,3...K is used at most once( ie a number can be used one or 0 times) Also we need to ...
2
votes
1answer
31 views

compute $\sum_{i=0}^{\infty}(DY)^i$ from $\sum_{i=0}^{\infty}Y^i$

I have an $m \times m$ matrix $Y$ , and an $m \times m$ diagonal matrix $D$. Now suppose that result of the matrix $X=\sum_{i=0}^{\infty}Y^i$ is given, and I want to compute the matrix ...
1
vote
1answer
292 views

Finding determinant for characteristic polynomial

The question I'm currently working on has boiled down to $\chi_A(t) = \det \begin{bmatrix} t & 0 & 0 & \cdots & 0 & -a_0 \\ -1 & t & 0 & \cdots & 0 & -a_1 \\ ...
2
votes
2answers
98 views

Eigen values of real symmetric matrix to occur in signed pairs

What condition, if any, can be said about a real symmetric matrix to have all its eigen values appear in pairs with opposite sign but same magnitude, i.e. if $\lambda$ is an eigen value of A then ...
2
votes
1answer
155 views

Simple proof involving eigenvectors and eigenvalues

I have the following question: Let $T$ be a linear operator on a finite dimensional vector space for which every non-zero vector is an eigenvector. Prove that $T$ is a multiplication by a ...
1
vote
3answers
131 views

conditions under which real-matrix exponential are equivalent

Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$? Thanks!
2
votes
2answers
368 views

Equivalence classes of similar $2\times 2$ matrices

How can we describe the equivalence classes under the similarity relation for $2 \times 2$ matrices with respect to the field of real numbers, $\mathbb{R}$? How would the equivalence classes change if ...
0
votes
1answer
373 views

Minimizing a function using gradient (example from Wikipedia)

This example is from Wikipedia (http://en.wikipedia.org/wiki/Gradient): The gradient of function $f(x,y,z)=2x+3y^2-sin(z)$ is $\nabla f= \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial ...
6
votes
1answer
161 views

Relate the singular values of $A$ and $\frac{A^T+A}{2}$

Consider a square matrix with real entries $A\in\mathbb{R}^{n\times n}$ such that the symmetric matrix $\frac{A^T+A}{2}$ is positive definite. Is it possible to find a relationship linking together ...
4
votes
5answers
216 views

Linear Algebra - Rank of a matrix

A is a $100 \times 100$ matrix. The element in the $i^{th}$ row and $j^{th}$ column is given by $i^2 + j^2$ Find the rank
0
votes
1answer
55 views

Projecting two vectors to have constant element-wise euclidean distances

Let $x = (x_{0}, \ldots, x_{n})$ and $y = (y_{0}, \ldots, y_{n})$ be two vectors, and $d = (d_{0} = x_{0} - y_{0}, \ldots, d_{n} = x_{n} - y_{n})$ their element-wise euclidean distances. My question ...
0
votes
1answer
29 views

Easy question regarding quadratic equations

Note: This is part of my preparations for my exams, im getting the wrong answer as $32/17$ which is not what wolfram alpha says, i would highly appreciate it if somebody could provide a direct answer ...
0
votes
1answer
54 views

Upper Triangular Matrices of Monotone Vectors

I am looking for references to the following problem (I'm actually interested in general $n$, but will use $n=3$ as an example): consider a finite set, for example, N = {1,2,3}, and the associated ...
2
votes
2answers
72 views

Prove that $f_{\beta'} = (f_{\beta})^{-1}$

I'm stuck on this problem for few days and can't find the solution.Hope some one here can help me solve this. I'm so grateful for any any help: Let $V$ be a finite-dimensional vector space ...
0
votes
1answer
80 views

Bounding the smallest eigenvalue of an ergodic Markov Chain

I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
0
votes
5answers
142 views

Two easy questions for arithmetic progression

I was not able to solve either of these, I kept hitting many mistakes and it would be much appreciated if the solution to these two could be provided, thanks a lot in advance. If the sum of all the ...
2
votes
2answers
89 views

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…

Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$ So I've tried to put the equation of plane ...
1
vote
4answers
95 views

Find the equation of plane containing line described by

Please help me in this really easy task Find the equation of plane containing line described by $x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
4
votes
2answers
250 views

How fast can you determine if vectors are linearly independent?

Let us suppose you have $m$ real-valued vectors of length $n$ where $n \geq m$. How fast can you determine if they are linearly independent? In the case where $m = n$ one way to determine ...
1
vote
1answer
143 views

Size of linear independent set = dimension of vector space

Let $V$ be a vector space of dimension $dimV =n$, over a field F. $S$ is a linearly independent set. Suppose I am equipped with the fact that the size of any linearly independent set must be $\le$ the ...
1
vote
1answer
52 views

Generalization of zero-diagonal square matrices to linear operators

Which linear operators in Banach or Hilbert spaces (e.g., partial differential operators or some other operators in functional spaces) are generalizations of square matrices $A=(a_{ij})$ such that ...
2
votes
2answers
329 views

Eigenvalue & Eigenvector question

So far, I have part (A) done of this question, but I am REALLY stumped at part b and c. Can anyone perhaps help me out? The city of Mawtookit maintains a constant population 300,000 people from year ...
1
vote
1answer
81 views

Is it possible for these to be onto?

Is it possible for a linear map to be onto if: The domain is $R^5$ and the range is $R^4$? The domain is $R^5$ and the range is $M(4,4)$? The domain is $R^5$ and the range is $F(R)$? I know to be ...
2
votes
3answers
135 views

Diagonalizing Matrices

Show that if $A$ and $B$ are two $n \times n$ matrices that both have the same diagonalizing matrix $X$, then $AB = BA$ I have the following answer, I just don't understand it completely. Can someone ...
1
vote
3answers
363 views

Steinitz exchange lemma

How can I show this? if $b_{1}, ..., b_{n+1}$ are linears combinations of $a_{1}, ..., a_{n}$ then $b_{1}, ..., b_{n+1}$ are linearly dependents. In my textbook they call it Steinitz lemma. I wonder ...
2
votes
3answers
279 views

Is the sum or product of idempotent matrices idempotent?

If you have two idempotent matrices $A$ and $B$, is $A+B$ an idempotent matrix? Also, is $AB$ an idempotent Matrix? If both are true, Can I see the proof? I am completley lost in how to prove both ...
2
votes
1answer
604 views

Interpolation of surface normals on the face of a triangle and Goroud shading

I am learning lightning for 3D graphics. What I'm about to ask about is described on this tutorial page here. The color of a certain pixel with lightning is given by $$D*I*(\hat{L}\cdot \hat{N}))$$ ...
1
vote
2answers
77 views

Prove an inequality in a group ring

Let $$G=\bigoplus_{n\in\mathbb{Z}}\left(\mathbb{Z}/2\mathbb{Z}\right)_n$$ be a group, and for any $n\in \mathbb{Z}$, denote $\delta_n$ to be the element in $G$ with $n$-th coordinate $1$ and zero at ...
2
votes
2answers
2k views

An eigenvector is a non-zero vector such that…

Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...