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 ...

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most general form of $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution $X = f(A,B,C)$?

What is the most general form of the cubic matrix equation $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution of the form $X = f(A,B,C)$, where $A,B$ and $C$ are positive definite ...
2
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1answer
44 views

Positive definite martix

I understand the majority of this solution, it's just I don't understand why I have to use both $\epsilon_1 $ and $\epsilon_2 $ rather than just $\epsilon$. I understand that i'm working with ...
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0answers
20 views

Power iteration sequemce for a special nonnegative irreducible imprimitive matrix

Let $A \in \mathbb{R}^{n \times n}$ be nonnegative irreducible matrix with maximum positive eigenvalue equal to 1. Let's assume $A$ has $h$, $h > 1$ eigenvalues $\lambda_1, \dots, \lambda_h$ with ...
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3answers
68 views

Determinants Proof

Let A and B be square matrices. Prove (or disprove) the following $$\det(qA) = q^{n} \det(A).$$ I tried disproving it with counterexamples but I could not find one.
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0answers
21 views

find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes

My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation: $x^T U \Sigma^2 U^T x = 1$ I know the lengths of it's principal axes (therefore the $\Sigma$ ...
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3answers
41 views

Question about eigenvalue and determinant

Question : If $A$ is an $n \times n$ matrix with real entries and $n$ eignvalues $(a_{1},a_{2},a_{3},a_{4},\cdots,a_{n})$, then does $det|A|=a_{1}a_{2}a_{3}a_{4}\cdots a_{n}$? (THE ANSWER IS : YES) ...
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1answer
39 views

Determinant of Identity minus a singular matrix

I am interested in calculating, or bounding in some way, the following determinant \begin{equation} \det\left[\mathcal{I}-Rxx^t\right] \end{equation} Here, $Rxx^t$ is clearly a singular matrix. Im ...
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0answers
39 views

diagonal of pseudoinverse of laplacian matrix

I have to find the diagonal of the pseudoinverse of a laplacian matrix evaluated on a directed and weighted graph. My laplacian is defined as: L = D - A where: D is a diagonal matrix; Di,i the sum ...
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1answer
28 views

How the Wronskian works

To prove linear independence of a set of functions, we say that given their Wronskian matrix W, Wx = 0 implies trivial solution (0,0,0,...) if the value (determinant) of the Wronskian is identically ...
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1answer
22 views

Linear Algebra - elimination and linear systems

By given this matrix: \begin{pmatrix}1&1&1&0\\2&3&k&1\\3&k&5&1\end{pmatrix} I need to find, what are the values of k the system has infinity/single/no solution. So ...
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34 views

Prove that the elements of these two sequences are not null

Let $x_{n+1}=x_n+2y_n$ and $y_{n+1}=y_n-x_n$, where $x_1=1$ and $y_1=-1$. I tried proving by contradiction, I tried by induction, I got nothing. This is a question I had on an exam, I didn't manage ...
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7answers
184 views

If $A^2 = B^2$, then $A=B$ or $A=-B$

Let $A_{n\times n},B_{n\times n}$ be square matrices with $n \geq 2$. If $A^2 = B^2$, then $A=B$ or $A=-B$. This is wrong but I don't see why. Do you have any counterexample?
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1answer
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why is algebraic multiplicity always equal to the geometric multiplicity of distinct eigen values corresponding to Symmetric matrices?

In other words why is symmetric matrix always diagonalizable? could someone explain intuitively?
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81 views

Caracterization of isometries that preserve time-orientation in $\Bbb L^3$

First of all, I'm considering $\Bbb L^3$ with the convention: $$\langle (x_1,y_1,z_1),(x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ Let $\Lambda = (\lambda_{ij})$ be an isometry of $\Bbb L^3$. I ...
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2answers
33 views

Matrix Algebra, Signs of solution

I have a system $AX = B$, where $A$, $B$ and $X$ are $N \times N$ matrices. I am interested in the properties of the solution $X$. $B$ has the following property: the diagonal terms are strictly ...
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0answers
25 views

Find the angle of rotation about a vector caused by application of a rotation matrix

I have a rotation matrix $R$ and a unit vector $\mathbf{v}$. How can I find the angle of rotation about $\mathbf{v}$ caused by the application of $R$?
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27 views

Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
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1answer
20 views

A question in matrix theory, SVD related.

For four $m\times n$ matrices A, B, A', B'. If $AA^\dagger=A'A'^\dagger, BB^\dagger=B'B'^\dagger$ and $AB^\dagger=A'B'^\dagger$, then if there always exists an unitary matrix V in U(n) such that ...
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2answers
36 views

Least Squares Solution Confusion

Say if I have an overdetermined system $A\vec x=\vec b$, I can use the normal equations $\implies$ $A^TA\vec x=A^T\vec b$. If I solve for $\vec x$ I will get a "solution" with an error. It says in ...
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2answers
35 views

Proving a subset is a subspace of a Vector Space

To prove a subset is a subspace of a vector space we have to prove that the same operations (closed under vector addition and closed under scalar multiplication) on the Vector space apply to the ...
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1answer
9 views

Incremental Cartensian Coordinates Betwwen Two Known Coordinates

I've done a lot of searches and haven't found exactly what I'm looking for. I'm looking for an algorithm that will provide me the cartesian coordinates (xyz) every 100ft between two known cartesian ...
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1answer
86 views

Relationship between eigenvectors of matrices

I am investigating parameter estimation in reduced-rank regression and have come across the following linear algebra result which I haven`t been able to prove. Suppose, $A \in \mathbb{R}^{nxm}$ of ...
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1answer
29 views

Minimze min max (A*x)

has this example matrix A some special propertries, which might be useful? $$ \left[\begin{array}{rrrrrr} 3 & 0 & 0 & 0 & 2 & 0 & 0 & 0 \\ 4 & 3 & 0 & ...
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3answers
82 views

Proof needed for this exercise from “Linear Algebra Done Right”

Suppose that $U$ and $V$ are finite-dimensional vector spaces and that $S\in \mathcal{L}(V,W)$ and $T\in \mathcal{L}(U,V)$, where $\mathcal{L}(X,Y)$ is the vector space of linear transformations from ...
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0answers
44 views

basis exchange linear equation

If you have: $Ax=b$ If you apply a basis change to $A$ and to $b$. Is then the solution(s) $x$ the same? If $A$ is a sparse Matrix is then $A$ still a sparse Matrix?
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2answers
27 views

Differentiation of polynomial as a linear map

Define D: P$_{2}$($\mathbb{R}$) $\mapsto$P$_{2}$($\mathbb{R})$ by D(p)(x) = p'(x) , Show D is linear? . Im a little unsure why this relation is linear?, if we let p(x) = ax$^{2}$+bx+c then p'(x) = ...
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3answers
41 views

How to determine for which value of an unknown parameter, one eigenvalue is 0?

Given the matrix: $A = \begin{bmatrix} a & 1 & 0 \\ 4 & a & 1 \\ 0 & 0 & a \end{bmatrix}$ for which value of the parameter $a$ one eigenvalue is certainly equal to $0$? ...
2
votes
4answers
550 views

When does $Ax=b$ have any solutions?

Suppose $A$ is a $3\times 3$ matrix with columns $v_{1}, v_{2}, v_{3}$. If $b = 2v_{1}- v_{3}$, then $Ax = b$ has one or more solutions. Is this true or false? Is this false, since $Ax= b$ has ...
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1answer
41 views

Calculate Points Between Two Points

I have two points $(A, B)$, both with longitude and latitude. For each point I have a speed in $km/h$, I assume a car drives from point $A$ to point $B$. I already have a function to calculate the ...
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0answers
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How to import matrix from sif document

I want to make some computation (with scilab, scipy or other) over the matrix A of linear problems (in inequational form). Those problems are in .sif format (from the netlib library in fact) and I ...
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1answer
26 views

Prove Basis for symmetric matrix.

**Let V be the vector subspace of M$_{2}$ ($\mathbb{R})$ consisting of all symmetric matrices, That is A$^{t}$ = A. 1) Show that $\clubsuit$= $\left\{ \left(\begin{array}{cc} 1 & -2\\ -2 & 1 ...
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1answer
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$m=17 \cdot 23 = 391$. With an exponent $e=3$ and encrypted word is $c=21$. Decrpyting exponent $d=235$. Find $w$.

Say $m=17 \cdot 23 = 391$. With an exponent $e=3$ and encrypted word is $c=21$. Decrpyting exponent $d=235$. Find $w$, when $w \equiv c^{d} \pmod{m}$. So far I have split it up like this: ...
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2answers
55 views

Why is tensor product of linear maps defined as $(S\otimes T)(v\otimes w)=S(v)\otimes T(w)$?

In my understanding, the definition of tensor product of linear maps cannot be directly derived from the definition of tensor product of vector spaces (or modules), since it's not clear what is the ...
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1answer
31 views

Finding linear map given a condition.

Given that $T:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is linear and $T(3,2) =(4,6)$ and $T (2,3) =(1,-1)$ ,how can I find $T (4,3)$ ?.
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22 views

Nonlinear Maps with additivity or homogeneity

Examples of linear maps from $\phi :R^2 \to R$ that has homogeneity but is not linear. Example of a function $\phi : C \to C$ that is additive but is not linear. All the examples I have found for ...
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1answer
32 views

Finding and proving a basis for $W=\{f(x) \in P_2[\mathbb{R} ]:f'(x) +xf(0) = 0 \}$

I'm having a trouble proving/finding a basis for $W= \{f(x) \in P_2[\mathbb{R}]:f'(x) +x \bullet f(0) = 0 \}$. I'm supposing $\{ x, 1 \}$ is a basis for W because any vector in $P_2[\mathbb{R}]$ gets ...
0
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1answer
55 views

Norm of a Vector

Suppose $A\inℝ^{n,n}$. We Define $$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$ Show it is a norm. Any thoughts?
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1answer
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Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
3
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1answer
43 views

Do invariant lines of linear transformations contain a fixed point?

Suppose $A$ is a $2$-by-$2$ matrix, and $\mathcal{l}$ is an invariant line under $A$, so $(x,mx+c)$ is mapped to $(X,mX+c)$ for some variable $X$ linear in $x$. Then is there a point on the line ...
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4answers
65 views

Find the basis for the subspace of the set of polynomials of degree less than five?

Let U = {p $\in P_4(F): p(2) = p(5) = p(6)$. Find a basis for U. I know how to do this problem if I were given p(2) = p(5). Set the two equal to each other and solve for one of the coefficients. I ...
3
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1answer
92 views

Is it true that if $A=BA^{*}A$ then $A^{*}=B^{*}AA^{*}$

I wonder is it true that for any $n \times n$ matrices $A$, and $B$.// If $A=BA^{*}A$ is true, does it imply that $A^{*}=B^{*}AA^{*}$?// I used mathematica to check the condition in 3 by 3 case, and ...
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0answers
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Canonical term for $\overline X / X$ where $X$ is a normed space.

Let $X$ be a normed vector space. Let $\overline X$ denote its completion. Is there a canonical name for the quotient space $\overline X / X$? Some authors seem to use "torsion" as a name, but I ...
2
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1answer
62 views

Which Subspaces do Antisymmetric Tensors Represent?

So antisymmetric tensors represent volumetric subspaces (I've asked this here instead of on phys.stackexchange because it seems like more of a math question)? How exactly would one know WHICH ...
0
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2answers
80 views

Which Linear Algebra textbook would be best for beginners? (Strang, Lay, Poole)

I am looking at buying 1 of the 3 following Linear Algebra texts for my reference. Introduction to Linear Algebra by Gilbert Strang 4th edition Linear Algebra and its Applications by David Lay 4th ...
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0answers
19 views

Detecting coplanarity by given pairwise distances

Given a 3D point set $P$, where $|P| \gg 4$ is there a way other than using Cayley-Menger determinant to detect if a group of points are coplanar or not? In other words, what are the methods to ...
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Find an intersection of a ray with a surface [closed]

I want to check if a ray hits a surface. In that case the surface is a coons patch. What is the way to do this?
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2answers
77 views

How to find the determinant of this matrix

I'd like to find the determinant of following matrix $$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & ...
2
votes
1answer
47 views

interger matrix whose square is identity

how can we find all the matrices with integer entries of size $n \times n$ such that $A^{2}=I$ and the matrix does not have fixed point in $\mathbb{Z}^n$ (except zero of course)? $-I$ is one example. ...
0
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2answers
35 views

map between finite and infinite vector spaces

I am sorry for this basic question: let We have two vector spaces such that one of them is finite dimensional and another one is infinte dimensional. I want to know whether I can define a linear map ...
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3answers
194 views

How prove $A^2=0$,if $AB-BA=A$

let $A_{2\times 2}$ matrix, and The matrix $B$ is order square,such $$AB-BA=A$$ show that $$A^2=0$$ My idea: since $$Tr(AB)=Tr(BA)$$ so $$Tr(A)=Tr(AB-BA)=Tr(AB)-Tr(BA)=0$$ Question:2 if ...