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
2answers
19 views

Choose $h$ and $k$ such that the system has, no solution, a unique solution, and many solutions.

Looking through my textbook, I see no examples as to how to solve this \begin{align} x - 3y & = 1 \\ 2x + hy & = k \end{align}
0
votes
1answer
14 views

Area of a triangle - straight lines

Question: What is the area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^2 - y^2 + 2y = 1$. Well I really have no idea how to even start the ...
1
vote
1answer
32 views

Finding minimum point of a function using linear algebra

Given a function $$q(x,y)=2x^2-2xy +2y^2$$. Find the minimum point of the following function by first converting it to a matrix form and using the diagonalisation of the matrix to find its minimum ...
2
votes
2answers
36 views

Question about inner products

Given a real or complex vector space $\;V\;$ and a (finite) basis $\;B\;$ of it, does it always exist an inner product on $\;V\;$ s.t. $\;B\;$ is an orthonormal basis with respect to it? The question ...
0
votes
0answers
11 views

Group inverse of positive semi definite matrix

Group inverse and Moore Penrose inverse of a positive semidefinite matrix are same. How?
1
vote
2answers
44 views

For What $a$ The Linear Equations Have Sloutions

$2x+ay-z=-2$ $x-3z=-3$ $x+2y+az=-1$ I have thought about reducing a matrix so in the end I will have an equation with $a$ then I can determine for which $a$ the are one solution/infinite ...
0
votes
0answers
12 views

Minimization using Davies, Swann and Campey algorithm [on hold]

How to use the Davies, Swann and Campey algorithm for minimizing a function of 2 variables?
1
vote
0answers
14 views

Trick for Jordan-Matrix and transformation of basis

some time ago I found a 'trick' for getting a basis-transformation-matrix for jordan. I'd like to understand it, but at a certain point I stuck. Maybe you can help me? Given is a matrix A: ...
2
votes
0answers
13 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
2
votes
2answers
54 views

If a linear transformation $T$ is nilpotent, show that $\alpha_0+\alpha_1T+…+\alpha_kT^k$ is invertible

If a linear transformation $T$ is nilpotent, show that $\alpha_0+\alpha_1T+......+\alpha_kT^k$ is invertible provided that $0\ne\alpha_0\in F,$ for some field $F$. I am in the mid way, and am stuck at ...
0
votes
4answers
50 views

System of linear equation

Determine the value for k for which the system of linear equation has infinitely many solution. \begin{cases} 2x - y = 2\\ 4x + ky = 4 \end{cases}
0
votes
1answer
27 views

Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
0
votes
1answer
19 views

Problem book for abstract linear algebra

Kindly suggest a good book for abstract linear algebra other than finite dimensional vector space by P R Halmos
1
vote
0answers
7 views

Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
0
votes
2answers
18 views

Show that every row of matrix $S$ is a linear combination of its bottom row and the row (1 1 1 1 1 1 )

Couldn't solve the following three questions. $$S=\begin{pmatrix} 36 & 35 & 34 &33&32&31 \\ 25 & 26 & 27&28&29&30 \\ ...
0
votes
1answer
13 views

inequality of Kernels dimension

Exercise Let $U,V,W$ be $K$-finite-dimensional vector spaces, and $f \in \operatorname{Hom}_K(U,V)$, $g \in \operatorname{Hom}_K(V,W)$. Show that $\dim(\ker(g \circ f))\leq ...
1
vote
1answer
30 views

Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?

Here's a homework question I've been stuck on for a while. My question is what can you tell about $$\lim_{t\rightarrow\infty}e^{tA}$$ if A is $n\times n$ matrix and you know that every eigenvalue of A ...
0
votes
2answers
25 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
0
votes
1answer
13 views

Problem related to dual space of infinite dimensional v.space $V$

Let $V$ be a $K$-infinite dimensional vector space, and let $\mathcal B$ be a basis of $V$. For each $v \in \mathcal B$, let $\phi_v \in V^*$ given by $\phi_v(v)=1$ and $\phi_v(w)=0$, for all $w \in ...
0
votes
1answer
19 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
12 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
0answers
7 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 ?
0
votes
3answers
52 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 $\mathbb{C}$ ?
-3
votes
0answers
20 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
0answers
28 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
32 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
2answers
28 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 ...
-5
votes
0answers
42 views

How does an elliptic element of $\mathrm{SL}(2,R)$ conjugates to a rotation? [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 $|\mathrm{tr}(A)|< 2$. ...
0
votes
1answer
19 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 ...
0
votes
0answers
17 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
1answer
42 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 ...
0
votes
2answers
31 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
19 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} & ...
3
votes
0answers
15 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
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
2answers
34 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} ...
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 ...
1
vote
3answers
41 views

Span of columns (or rows) of a given 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 $\mathbb{R}^3$ ...
3
votes
2answers
28 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 = ...
4
votes
1answer
66 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 ...
2
votes
1answer
29 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
38 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 + ...
-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 ...
0
votes
2answers
49 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
0answers
16 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} ...
0
votes
1answer
46 views
0
votes
1answer
29 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
27 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.
1
vote
1answer
19 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 ...
1
vote
2answers
45 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 ...