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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

20
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3answers
9k views

Integer matrices with integer inverses

If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it's not ...
3
votes
1answer
165 views

Linear Systems: Elimination

I am taking a beginner linear algebra class and would like a hint on a homework question. I don't need the answer, just a little guidance. Consider the differential equation: $$\frac{d^2x}{dt^2} - ...
3
votes
1answer
241 views

Inverting $(A + D_p)$ efficiently for many diagonal $D_p$ (and $A$ SPD)

At one step of an algorithm (i=1: something big) I have to draw a vector from the normal distribution $X_p^{(i)}\sim N((A^{(i)} + D^{(i)}_p)^{-1}m, (A^{(i)} + D^{(i)}_p)^{-1})$ for $p=1:P$, with a ...
5
votes
4answers
3k views

Spectrum of the sum of two commuting matrices

Suppose $A$ and $B$ are two commuting square matrices over an algebraically closed field. It is true that the spectrum of $A+B$ is contained in the set $\{\lambda_1+\lambda_2:\lambda_1 \in \sigma(A), \...
2
votes
0answers
387 views

From a triangular matrix to a symmetric square matrix

I am a computational physics postgrad student, working with libraries like ATLAS and MAGMA. I have a matrix which is upper-triangular, and is the result of a Cholesky decomposition. I need to ...
5
votes
3answers
676 views

Is there any relation between the eigenvalues of possibly non-Hermitian matrix A and those of exp(A)?

Is there any relation between the eigenvalues of possibly non-Hermitian matrix A and those of exp(A)? For hermitian matrices, they are just exponentials of the corresponding values. But in general, ...
1
vote
2answers
386 views

Find a T from $R^3$ to $R^4$ given an equation for a subspace in $R^4$

A subspace V in $R^4$ is defined by the equation $x_{1}$-$x_{2}$+$2x_{3}$+$4x_{4}$=0. I need to find T such that Ker(T)=zero vector, and Im(T)=V. How do I approach this problem? As I understand, the ...
2
votes
3answers
2k views

If Ker(A)=$\{\vec{0}\}$ and Ker(B)=$\{\vec{0}\}$ Ker(AB)=?

I need to demonstrate that Ker(AB)=${\vec{0}}$, just like Ker(A) and Ker(B). I have a not-so-elegant way of showing that (I think): A and B are n * p and p * m, and it has to be the case that n $\geq$ ...
8
votes
1answer
2k views

connection between graphs and the eigenvectors of their matrix representation

I am trying to learn graph theory and the linear algebra used to analyse graphs. The texts I have read through have lots of lemmas and theorems proved. The proofs are convincing but I fail to see the ...
2
votes
1answer
191 views

What is the locus such that any vector from it has a given dot product with the given vector?

Consider a given vector $a$ and scalar $d$. What is the set $X$ such that for any $x \in X$ their dot product equals $d$ : $\forall x \in X: x \cdot a = d$ ?
1
vote
1answer
3k views

Question about the standard basis's relationship to the identity matrix

I'm in the process of reading my first Linear Algebra textbook, and was just wondering...Is the standard basis of a vector space in n dimensions equivalent to the row space of the n x n identity ...
3
votes
1answer
370 views

Finding displacement from origin of SVM hyperplane

I'm working through Stanford's computer vision course to refresh some of my math skills (http://vision.stanford.edu/teaching/cs223b/syllabus.html), and I've run into a problem while working with a ...
11
votes
4answers
3k views

What about $GL(n,\mathbb C)$? Is it open, dense in $M(n,\mathbb C)$?

What about $GL(n,\mathbb C)$? Is it open, dense in $M(n,\mathbb C)$?
21
votes
2answers
8k views

Understanding the properties and use of the Laplacian matrix (and its norm)

I am reading the wikipedia article on the Laplacian matrix: http://en.wikipedia.org/wiki/Laplacian_matrix I don't understand what is the particular use of it; having the diagonals as the degree and ...
4
votes
4answers
13k views

Show that the linear transformation T is invertible

(Application of the rank-nullity theorem) Suppose $S,T: V\to V$ are linear transformations of a finite dimensional vector space $V$, and that the composition $ST\colon V\to V$ is invertible. Show that ...
4
votes
1answer
190 views

Why is this tuple of vectors not a basis for $\mathbb{R}^{2}$?

Why is the following tuple of vectors not a basis of $\mathbb{R}^{2}$? $\left ( \left ( \begin{array}{c} 1\\ 0\\0 \end{array} \right ), \left ( \begin{array}{c} 0\\ 1\\0 \end{array} \right ) \right )...
6
votes
1answer
885 views

1-dimensional solution space of homogeneous system Ax=0?

Given is an almost-square matrix $A$ with $n$ columns and $n-1$ rows with maximum rank. The solutions of the homogeneous system $Ax = 0$ form a 1-dimensional subspace of $\mathbb{R}^n$. I've ...
0
votes
1answer
685 views

Affine transformation matrix doing translation

Greetings All I have some test matlab code which can scale and do rotation but the translation (tl value) doesn't seem to be working. I expected the entire object to be moved over x=2 and y=0 but ...
2
votes
1answer
85 views

Help me spot the error?

I have a determinant to expand which is $$\triangle = \begin{bmatrix} p& 1 & \frac{-q}{2}{}\\ 1& 2 &-q \\ 2& 2 & 3 \end{bmatrix} = 0 $$ But when I am expanding the ...
1
vote
3answers
2k views

Given 5 equations, can I solve for 6 unknowns?

Background I am trying to extract data from scientific publications. Sometimes an experiment can test two factors, e.g. a standard two-way factorial ANOVA. Call these factors $A$ and $B$. If factor ...
3
votes
1answer
101 views

Behaviour of a numerical solution of an ODE

The following question was part of a problem sheet in numerical maths: Given the scalar equation $\ddot{y} = - \lambda y$, $\lambda > 0$ show that for $h > h_{critical}$, the approximate solution ...
1
vote
2answers
502 views

How to multiply vector and matrix expressions involving transposes

How do i expand and simplify the vector expression $(\vec{a}-\vec{b})^T.(\vec{a}-\vec{b})$ = ? And if there are matrices A and B instead of vectors a and b, how do i multiply and simplify this ...
11
votes
4answers
18k views

Linear independency before and after Linear Transformation

If we are given some linearly dependent vectors, would the T of those vectors necessarily be dependent (given a transformation from $R^n$ to $R^p$)? And if we are given some linearly independent ...
3
votes
1answer
653 views

Speeding up Gauss Elimination

I was working on something on Matlab couple of months back and found a tweak to speed up Gauss elimination on Matlab by dividing the original matrix into 4 block matrices and then solving them in a ...
1
vote
2answers
2k views

linearly independent Functions in linear algebra

I have this problem. I want to prove that if I'm choosing a basis for polynomials that their exponent is from $1$ until $n$ and they are all $F(2)=0$, they are linear independent. To be more spefic, ...
5
votes
1answer
2k views

Exterior power of dual space

Let $V$ be a vector space with basis $e_1, \ldots, e_n$ and $V^*$ be its dual space with dual basis $e_1^*, \ldots, e_n^*$. Let $k$ be an integer between $1$ and $n$. Why $\wedge^{n-k}V=\wedge^{k}V^*$?...
5
votes
2answers
541 views

Extending a positive linear functional in finite dimensions

Let $V$ be a vector subspace of $R^N$, and $l:V \to R$ a linear mapping such that $l(V\bigcap R_{+}^N)\subseteq R_{+}$ (i.e., $l$ is positive). I have heard that there exists a separating hyperplane ...
4
votes
3answers
273 views

proof that $\phi\circ\phi=id$ implies the existence of a diagonal matrix

As exam preparation we were trying to proof the following task: Let $V=\mathbb{R}^2$ and let $\phi$ be an endomorphism of $V$ with $\phi \circ \phi = id$ and $\phi \neq id$ and $\phi \neq -id$. Proof ...
4
votes
3answers
2k views

Diagonal of an inverse of a sparse matrix

Inverse of a sparse matrix could be dense, but what if I'm only interested in the main diagonal of the result? Is there a method that is more efficient than computing the full inverse?
2
votes
1answer
527 views

Linear isometry

Show that if $V$ is a finite-dimensional vector space with a dot product $\langle-,-\rangle$, and $f: V \rightarrow V$ linear with $\forall v,w \in V: \langle v,w \rangle=0 \Rightarrow \langle f(v),f(...
6
votes
2answers
357 views

If the product of an invertible symmetric matrix and some other matrix is symmetric, is that other matrix also symmetric?

The thought came from the following problem: Let $V$ be a Euclidean space. Let $T$ be an inner product on $V$. Let $f$ be a linear transformation $f:V \to V$ such that $T(x,f(y))=T(f(x),y)$ for $x,...
0
votes
1answer
179 views

Why doesn't the orthogonal trajectory maximize the Wronskian?

Drawing a parallel to linear Algebra. Premise: The maximum of a determinate is when all the columns are orthogonal, given that the vectors are normalized. Conclusion: The orthogonal trajectory ...
12
votes
3answers
7k views

'Free Vector Space' and 'Vector Space'

In this very nice book, author has defined vector space as set of functions $f : S \rightarrow F$ where $S$ is a finite set and $F$ is a field. It turns out that this definition is closely resemble ...
2
votes
1answer
267 views

A Projection Problem in Functional Analysis - Uniqueness of a Solution

I have the following system of equations: $$\alpha f_1(x) = \int_\mathbb{R} g(k) h_1(k) e^{\mathrm{i}kx} dk$$ $$\beta f_2(x) = \int_\mathbb{R} g(k) h_2(k) e^{\mathrm{i}kx} dk$$ with known functions ...
5
votes
1answer
2k views

Solving matrix equations of the form $XA = XB$

I am trying to solve the matrix equation of the form $XA = XB$. $A$, $B$ and the solution sought $X$ are $4 \times 4$ homegeneous matrices which are composed of a rotation matrix and translation ...
4
votes
1answer
239 views

What are the linear isometries on $R^n$, equipped with the $l_1$ norm?

Which conditions must the matrix entries satisfy, and what would be an interpretation of the row and column sums of the matrix?
2
votes
3answers
308 views

Combinatorial proof of $\Big|\prod\limits_{0 \leq i < j < N} (\zeta^j -\zeta^i)\Big| = \sqrt {N^N}$ for $\zeta \equiv \exp({2\pi i \over N})$

I've been playing with Fourier transform a little and discovered the identity quoted in the title. More precisely, writing the matrix for the Fourier transform in ${\mathbb Z} / N {\mathbb Z}$ as $$A ...
2
votes
3answers
246 views

Applying a linear transformation to time sequences to separate interfering oscillations

This is an applied problem, which arises from the problem of reorienting of a sensor axes according to particle displacement directions: Consider a sensor which is located inside the solid substance. ...
2
votes
1answer
104 views

A solution for equation with N unknowns with specific constraints?

I am working with granular materials (seeds). I am looking for a way to correctly scale the amount of different particles in one batch using weight only. I have worked with the problem a bit and ...
1
vote
1answer
3k views

How many presentable boolean functions with n attributes are linear separable?

The aim is to find a formula for the question. For n=2 i get (2^(2^n)=16 possible functions. This is the solution for a boolean function with 2 attributes: ...
4
votes
3answers
2k views

Decomposition of vector with respect to direct sum of vector subspaces

$U$ and $W$ are two subspaces of vector space $V$. If $U \oplus W = V$, then $\forall v \in V$, there exist two unique vectors $u \in U$ and $w \in W$ such that $v = u + w$. Is the reverse true? ...
5
votes
1answer
2k views

Cost of Solving Linear System

As most of us are aware the cost for solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{...
1
vote
2answers
396 views

Calculate the real eigenvalues and affiliated eigenspace of a square matrix whose elements are all 1

Question: Let $A \in M(n,n,\mathbb{R}), a_{ij} = 1$ for all $i,j$. Calculate the real eigenvalues and the affiliated eigenspace of $A$. So first of all what I would be trying to calculate are values ...
1
vote
3answers
2k views

Show that if A has a right-inverse, then $Ax = b$ has at least one solution for every choice of b in $R^n$

Show that if A has a right-inverse, then $Ax = b$ has at least one solution for every choice of b in $R^n$.
3
votes
3answers
247 views

Why is it that if A is an n by m matrix, and both BA and AB are indentity matrices, then A is square and can't be rectangular?

Some background: I was in class today, and the professor was proving that given a matrix $A$ that is $n$ by $m$ and a $B$ such that $AB=I$ where $B$ is obviously $m$ by $n$ and $I$ is $n$ by $n$, if ...
2
votes
1answer
138 views

Conditions on matrix expression

What conditions are needed on the matrices A and B so that the following is true: $A^{T}BA=B$. Clearly, if B is invertible, the determinant of A must be 1 or -1. What other conditions are needed?
3
votes
2answers
127 views

Suggest a tricky procedure

If $$A = \frac{1}{2} \times \begin{pmatrix} -1& -\sqrt3 & 0 \\ -\sqrt3& 1 &0 \\ 0 & 0 & 0 \end{pmatrix} \text{ and } E = \frac{1}{2} \times \begin{pmatrix} 1&...
13
votes
2answers
789 views

Matrices with real entries such that $(I -(AB-BA))^{n}=0$

I was just trying out some problems, when i couldn't solve this question: Does there exist $n \times n$ matrices $A$ and $B$ with real entries such that $$ \Bigl(I - (AB-BA)\Bigr)^{n}=0?$$ I really ...
5
votes
2answers
199 views

How many ways to choose $l$ vectors in $n$-dimensional space such that every $k$-subset is independent

Working in $F_q^n$. How many different ways do we have to choose $l$ vectors such that every subset of size $k$ of them is linearly independent. (Assume n is large) My Progress: For the first k ...
4
votes
1answer
2k views

transpose of positive matrix is positive

how to prove it? I am talking about matrixes which satisfy: $$( Ax , x ) > 0\quad \text{ for any}\quad \;x \neq 0.$$ How to prove that $A^T\;$ is also positive? $$x^T A x = ( x^T A x )^T$$ ...