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
18 views

Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
0
votes
1answer
10 views

exponential of a product of any two matrices commuting with one of the matrices

I'm trying to show that for any arbitrary matrices A and B, $e^{AB}A = Ae^{BA}$ I checked this other answered question, but this case differs as I have a product of matrices as opposed to a ...
0
votes
1answer
17 views

Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge 0$$ Where Q is square ($n$x$n$), positive semi ...
2
votes
0answers
45 views

Angle between two planes in four dimensions

Suppose I have two planes defined in 4D space, either in terms of vectors spanning the planes, $X = t_1 A_1 + t_2 B_2$ and $X = t_3 A_3 + t_4 B_4$ (where $X$, $A$'s, and $B$'s are vectors with four ...
0
votes
3answers
46 views

Matrices with $n$ eigenvalues [on hold]

My question is: how can I prove that the set of matrices with $n$ distinct eigenvalues is open in the space of $n\times n$ matrices over $\mathbb{C}$ ?
0
votes
0answers
5 views

Gram-Schmidt-procedure with PARI/GP

Can the Gram-Schmidt-procedure to find an orthogonal basis of a vector space spanned by given linear independent vectors be easily done in PARI /GP or do I have to program the procedure ?
-3
votes
0answers
18 views

Math issue implementing an invoice API [on hold]

Okay, so, I have $2$ separate systems: An invoice record database on an external site, I do not have access to the code here. An prestashop e-commerce installation, where i am developing a plugin. ...
0
votes
1answer
24 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
1
vote
1answer
24 views

Minimum eigenvalue of product of two matrices

Abstract description: Let $\mathbf{A}$ and $\mathbf{B}$ be two $n \times n$ real matrices. Let $\sigma( \mathbf{A B} )$ denote the spectrum of $\mathbf{A B}$. Assume that (A1) $\mathbf{A}$ is ...
2
votes
1answer
939 views

Orthogonal Projection of a Point onto a Plane

I'm dealing with an exercise that requires I find the orthogonal projection of a given point onto a given plane. I don't want an answer directly for my exercise, I would instead like to understand ...
0
votes
0answers
24 views

About lemma $\rho(A) \leq \|A^k\|^{1/k}$

In the Spectral radius wikipedia article in section Matrices there is a lemma, what states that: Lemma. Let $A \in \mathbb{C}^{n \times n}$ be a complex-valued matrix, $\rho(A)$ its spectral ...
0
votes
1answer
57 views
+50

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
1
vote
3answers
663 views

Finding a unique solution given rank $A = m$

So first off all, this WAS homework. Submitted it about 3 hours ago. Let $A$ be an $m \times m$ matrix. then show that if rank$ A = m$, then $Ax = 0$ has a unique solution. My roommate said this is ...
2
votes
3answers
3k views

How to build a linear equation system?

How can one build a $3\times 3$ linear equation system ($3$ equations with $3$ variables) if the variables are known, for example, $a=1, b=2$ and $c=3$? Thank you in advance!
1
vote
2answers
26 views

General Solution Of Linear Equations

$x_1+x_2-6x_3+4x_4=6$ $3x_1-x_2-6x_3-4x_4=2$ $2x_1+3x_2+9x_3+2x_4=6$ I have row reduced the matrix and got $$\left(\begin{array}{cccc|c} 1 & 1 & -6 & 4 &6\\ 0 & 1 & -3 ...
2
votes
1answer
136 views

When a system of rational linear equations have complex solutions does it have rational solutions?

Problem: When a finite system of rational linear homogeneous equations in finitely many variables have a nontrivial complex solution (that is not a rational solution), does it imply that there is ...
-5
votes
0answers
29 views

How does an elliptic element of $\mathrm{SL}(2,R)$ look like? [on hold]

Show that an elliptic element of $\mathrm{SL}(2,\Bbb R)$ is conjugate to a rotation, where an element $A \in \mathrm{SL}(2,\Bbb R)$ is called an elliptic element if $|\mathrm{tr}(A)|< 2$. ...
1
vote
1answer
49 views
+50

Is Frobenius norm induced up to a scalar factor?

I know that the Frobenius norm is not induced since $||I||_F=\sqrt n\neq 1$. But what if we consider the norm $\frac 1 {\sqrt n} ||\cdot ||_F$? Thank you!
2
votes
1answer
383 views

Reflection Matrix linear algebra

I am practicing some linear algebra question to prepare for my test. I have come across one question that has given me much trouble. It states: If $\lVert u\rVert = 1$, then $Q = I - 2uu^T$ is a ...
0
votes
2answers
60 views

Rotate secondary vanishing points to the primary vanishing points to find new length of object

all though only the 2D data is available, the best way to think of this problem is a piece of paper pinned at one corner to a wall, but the paper is sitting at an angle to the wall, see illustration ...
0
votes
0answers
15 views

How to attack solving for similarity transformed quantities

I'm interested in solving equations of the form: $$ R\mathbf{x}R^{T}=\alpha\mathbf{x}+k $$ where $R$ is a orthonormal matrix (rotation), $\alpha$ is a scalar multiplier (non-zero), ...
1
vote
2answers
920 views

Linear Combinations and solutions

Let A be a 5 x 3 matrix. If $$b = a_1 + a_2 = a_2 + a_3$$ then what can you conclude about the number of solutions of the linear system Ax = b? Explain. I'm not sure about this question. All I know ...
3
votes
2answers
26 views

Midpoint of chord of contact

Question: The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is: a) $20(x^2 - y^2)- 36x + 45y = ...
0
votes
1answer
25 views

How to find a general math formula of a vector and its matrix?

I have a vector x of size 1xM*N for some M and N. I ...
0
votes
0answers
16 views

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
2
votes
2answers
67 views

Geometric Product

I have a problem with the geometric product: In my book the unit trivector is defined like this: $(e_{1}e_{2})e_{3}=e_{1}e_{2}e_{3}$ But that would mean $(e_{1}e_{2})e_{3}= (e_{1} \wedge e_{2})\cdot ...
2
votes
0answers
6 views

Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
0
votes
2answers
31 views

Linearity In Linear Algebra

I am learning linear algebra for few months now and I came to the following notion. Due to the definition of field: $\sum_{i=1}^{n} \alpha(a_i+b_i)=\alpha\sum_{i=1}^{n} a_i+\alpha\sum_{i=1}^{n} ...
0
votes
0answers
22 views

Define a positive dot product in $\mathbb{R^3}$

Consider the matrix $A= \begin{bmatrix} k & k-1 & 0 \\ 1-k & 2-k & 0 \\ 2k-3 & 2k-1 & 2 \end{bmatrix} $ with $k \in \mathbb{R}$ and let be $f_a: \mathbb{R^3} \rightarrow ...
0
votes
0answers
20 views

How can I convert four dimensions into two?

I am trying to generalize the following problem (or at least extend it to 4 dimensions). If I have a 3 dimensional vector with the coordinates A,B,C and the constraints that A+B+C=1 and ...
0
votes
2answers
48 views

$A^{T}A$ is diagonal. What can I say about $A$?

Is there any special property about the elements of $A$ if $A^{T}A$ is diagonal? I imagine you need some sort of symmetry but I can't see what it should be. Edit: Sorry, maybe it's better phrased ...
2
votes
3answers
32 views

Spanning Matrix

Consider the following matrix: $A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \end{pmatrix}$ The columns of $A$ span $\mathbb{R}^2$. The columns of $A$ span $\mathbb{R}^3$ ...
0
votes
0answers
28 views

Least squares, plotting in Mathematica

According to Newton’s second law of motion, a body near the earth’s surface falls vertically downward according to the equation where vertical displacement downward relative to some fixed point. ...
1
vote
0answers
21 views

Proving a basis generates a vector space

I was given the following problem (this is actually homework, however, it seems like the tag is deprecated). Let $X \neq \phi$, $K$ be a field, then $K^X = \{f : X \to K\}$ is the set of all functions ...
2
votes
1answer
28 views

Surjective function from a set of funtions to itself

Let a function be defined by $f \longmapsto f'$ acting from the set of all polynomials to itself. I am asked if this is surjective. I would like to think it isn't, but I'm in doubt how I should ...
-2
votes
0answers
37 views

Straight lines - pair of lines [on hold]

Question: Let PQR be a right angled triangle with right angle at P(2, 1). If the equation of the line QR is $2x+y=3$, then the equation representing the pair of lines PQ and PR is: a) $3x^2 - 3y^2 + ...
3
votes
1answer
148 views

Vector space proof

Let $\mathbb F$ be a field and let $V$ be a vector spaces over $\mathbb F$. Show that for all $w\in V$, $(-1_{\mathbb F}) \cdot w=-w$, where $1_{\mathbb F}$ is the multiplicative identity of $\mathbb ...
13
votes
2answers
245 views
+50

Families of Idempotent $3\times 3$ Matrices

I did the following analysis for $2\times2$ real idempotent (i.e. $A^2=A$) matrices: $$ ...
1
vote
1answer
16 views

Showing unitary similarity of these two matrices

Let $A \in B(H)$ for a Hilbert space $H$, and $\alpha \in \sigma_{p}(A)$, the point spectrum of $A$. Suppose ker$(\alpha I-A)$ is not a reducing subspace of $A$ then $A$ has the following matrix ...
1
vote
2answers
44 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...
0
votes
1answer
43 views
-1
votes
1answer
16 views

Straight lines - point of intersection

Question: Two rays in the first quadrant: $$x +y = |a|$$ $$ax - y = 1$$ intersect each other in the interval $a \in (a_0, \infty)$, the what is the value of $a_0$? I don't even understand where to ...
6
votes
1answer
347 views

System of linear equations having a real solution has also a rational solution.

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $b ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $Ax = b$ has a solution in $\mathbb{R}^n$. Does it necessarily have ...
1
vote
2answers
114 views

Rational Solution of a System of Linear Equations [duplicate]

I am having a little trouble with this problem - Let $A$ be a $m\times n$ matrix and $v$ be a $n\times 1$ matrix, both of which only has rational entries. It is known that the equation $Ax=v$ has ...
1
vote
0answers
13 views

Exploiting structure in multilinear equations

I'm wondering if there are any standard techniques for exploiting structure in multilinear equations. An example of what I have in mind is solving $A_{ab} X_{bc} A_{cd} (B_{ad} B_{bc} + B_{ac} ...
1
vote
1answer
28 views

Norm of functional associated to vector $p$-norm [duplicate]

I read that the norm of a linear functional $f:V\to K$, with $K=\mathbb{R}\lor K=\mathbb{C}$, associated to the $p$-norm $\|x\|=(\sum_{i=1}^n|x_i|^p)^{\frac{1}{p}}$, for $p>1$, is ...
1
vote
1answer
14 views

Power of a matrix, given its jordan form

Can someone please explain how to find the power of a matrix $A$, given $A=MJM^{-1}$ where the matrix $J$ is in the Jordan canonical form? Or else please explain how to find the powers of a matrix ...
0
votes
1answer
27 views

What is the Singular Value Decomposition for the Zero Matrix?

I am interested in the singular value decomposition of a matrix: $\mathbf{M} = \mathbf{U} \mathbf{S} \mathbf{V}^T$. Suppose $\mathbf{M} = \mathbf{0}$ (zero matrix) and square. Clearly, $\mathbf{S} = ...
0
votes
2answers
26 views

Matrix in $\mathbb{Z}_5$

Let $A=\begin{bmatrix}3&2\\3&3\end{bmatrix} \in M_2(\mathbb{Z}_5).$ Then if I calculate $A^{105}$ like $105 \equiv 0 \pmod 5$ , $A^{105} = Id_2$ ? Thank you.
8
votes
2answers
251 views

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...