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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.

2
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1answer
51 views

Easiest way to show positive semi-definite equivalence

For an $x \in \mathbb{R}^n$, and $n$-by-$n$ identity matrix $I_n$, we are given that $$ \begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix} \succeq 0.$$ What is the easiest way to show that $$ \...
1
vote
1answer
54 views

$XA=A^TX$ prove $X$ symmetric matrix

Let $A$ be a nonderogatory matrix. This means the characteristic polynomial and the minimal polynomial of $A$ are coincide. Or, equivalently, every matrix $X$ that satisfies $XA=AX$ can be written as ...
1
vote
2answers
58 views

An explicit “formula” for the prime counting function?

It is known that $\log(p_1),\cdots,\log(p_n)$ are linearly independet over $\mathbb{Q}$, where $p_i$ denotes the $i$-th prime. For a number $1 \le k \le n$ let $Log(k)$ denote the vector with respect ...
0
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1answer
54 views

Complex transformation that transform square into a circle.

Bonjour. I’m looking for a conformal mapping that transform a square into a circle, a cube into a sphere, eventually a rectangle into an ellipse.
1
vote
1answer
20 views

Determine all positive powers.

Determine all positive powers of the $5 \times 5$ matrix... $$C=\begin{bmatrix} 0&0&0&0&0 \\ 1&0&0&0&0 \\ 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&...
0
votes
1answer
12 views

Suppose a square matrix $A$ has spectral radius $\rho(A) < 1$. Fixing the last row and scaling other entries by $r \in (0,1)$, will $\rho(A)<1$?

Suppose $A \in M_n(\mathbb R)$ has spectral radius small than $1$, i.e., $\rho(A) < 1$. Denote $A = \pmatrix{a_1^T \\ \vdots \\ a_n^T}$, where $a_j^T$ denotes the $j^{th}$ row of $A$.Putting $B=\...
1
vote
2answers
47 views

Show that any $M\in\mathcal{M}_n(\mathbb{C})$ of rank $r$ can be written as $M = A N_r B$

Let $\mathcal{M}_n(\mathbb{C})$ be the set of all $n\times n$ matrices with entries in $\mathbb{C}$ and $\mbox{GL}_n(\mathbb{C})$ denote the set of invertible $n\times n$ matrix in $\mathcal{M}_n(\...
0
votes
0answers
7 views

Change of basis of a lattice

Let $\Gamma:=\Big\{(i, 0), (0,i), (\sqrt 3,\sqrt 3)\Big\}$ be a basis of a lattice in $\mathbb C^2$. Can we find another basis $\Gamma':=\{(1,0),(0,1), w\}$ such that span over $\mathbb R$ of $\Gamma$...
1
vote
2answers
589 views

The number of scalar additions required to compute P(QR)

Let P, Q, R be matrices of order $3\times5, 5\times7$ and $7\times3$ respectively. What is the number of scalar additions requiered to compute P(QR)? Do we have any formula to compute this? If we had ...
0
votes
0answers
15 views

The definition of normal bundle as a quotient.

Let $f: N \to M$ be a smooth immersion and let $p \in M$, $W = f(V) \subset M$ be an submanifold with $q = f(p).$ Then the sequence is split exact $$T_qW \hookrightarrow T_qM \stackrel{\mu}\to T_qM/...
2
votes
2answers
545 views

Method to check for Positive definite matrices.

I think its already been asked , but still i can't figure out a way to do it computationally, I had to check for positive definiteness of a matrix $A$ of order $n$ by $n$.I know that for any vector $...
0
votes
2answers
35 views

How can we generalize the fact of finite dimensional vector space to an infinte dimensional case?

I am reading vector space from Friedberg. There in the last section they told about infinite dimensional vector space but there is not sufficient contents. Now my question is why can't we define ...
0
votes
3answers
49 views

If $A$ is a $3\times3$ matrix, then $A-A^2 \neq I$

Suppose a matrix $A \in M_{3\times3}(\mathbb R)$, then $A-A^2 \neq I$. I know that I should contradict that statement, and use the fact that a $3\times3$ matrix has at least one real eigenvalue. ...
0
votes
0answers
9 views

Describe vector subspaces and FInd basis for sum of vector subspaces

I need help with this problem: For part a), I thought to write $v_1,v_2,v_3$ in their vector form with respect to the standard basis. Then I will group them together into a matrix and row reduce the ...
1
vote
1answer
30 views

Rotation of Linear Transformation in R2

I have a matrix which represents a 45 degrees rotation counterclockwise: $$\begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$. ...
2
votes
1answer
17 views

How to prove $\text{det}(I+xy^{\top}+wz^{\top})=(1+y^{\top}x)(1+z^{\top}w)-(x^{\top}z)(y^{\top}w)$?

Suppose $x,y,z,w$ are vectors in $\mathbb{R}^n$ and $I$ is the identity matrix. Show that $\text{det}(I+xy^{\top}+wz^{\top})=(1+y^{\top}x)(1+z^{\top}w)-(x^{\top}z)(y^{\top}w)$.
2
votes
3answers
64 views

Intuition of generalized eigenvector.

I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...
0
votes
0answers
6 views

Jacobi matrix with complex entries and optimization's orientation

What can we tell about the direction/ orientation of an optimization if the entries of the Nx1 Jacobi-matrix (gradiant) are complex? According to wikipedia: "if the Jacobian determinant at p is ...
1
vote
2answers
38 views

How to Find limits and co-limits of diagrams over Vector?

I am having trouble understanding how to find limits and colimits of specific diagrams over the category of finite dimension vector spaces. I understand the definitions of cones, terminal objects, ...
1
vote
2answers
23 views

If $A$ is an anti-symmetric matrix of size $N$ where $N$ is odd, then $\lambda=0$ is an eigenvalue of $A$

True or False: If $A$ is an anti-symmetric matrix of size $N$ where $N$ is odd then $\lambda=0$ is an eigenvalue of $A$. Through observation, it is false. However, how can I prove that this statement ...
8
votes
2answers
123 views

If $B$ is nilpotent and $AB=BA$ then $\det(A+B) = \det(A)$

The following stumps me: Let $\mathbb K$ be a field. Let $A, B \in \mathbb K^{n \times n}$ where $B$ is nilpotent and commutes with $A$, i.e., $A B = B A$. Show that $$ \det(A+B)=\det(A) $$ I have ...
1
vote
0answers
15 views

Why this formulas is ill-conditioned and require a sufficiently high precision

The $L_2^\star$ discrepancy ($T^\star$) is a measure of the uniformity of N terms of random or quasi-random sequences. I read in a research paper that this quantity measure ($T^\star$) is ill-...
2
votes
2answers
762 views

Show that if $T$ is surjective and spans $V$, then $T(S)$ spans $W$.

Given that $T: V \to W$ is a linear transformation from $V$ to $W$. Show that if $T$ is surjective and $S\subset V$ spans $V$, then $T(S)$ spans $W$. I think the main thing stumping me right now is ...
0
votes
2answers
558 views

How to show a set spans a space?

I've just started working with abstract algebra, and while the theory makes some sense, I have a bit of trouble figuring out the actual methods to complement the theory. For example, a base for a ...
0
votes
1answer
22 views

How to show that vector $b$ is in the vector space $V$?

How do I show that vector $b$ is in the vector space $V$? vector $b = (0, 4, 7)$, $V = \operatorname{span} \{(1, 2, 2), (1, 1, 1), (-1, 0, 1)\}$ Do I use the equation $b = c_1 v_1 +c_2 v_2 +c_3 v_3$...
0
votes
1answer
39 views

Matrices: help with homework

I need to prove that $\vec{x}$ is a solution of $A\vec{x}=\vec{b}$: $$ \begin{vmatrix} 2&-7&-3\\ -4&1&5\\ 1&3&-1\\ \end{vmatrix} \cdot \begin{vmatrix} 5\\ -1\\ 7\\ \end{...
0
votes
2answers
27 views

Get rectangular coordinates of a 3d point, with the polar coordinates?

Ok, so imagine a cannon on an cartesian space. X and Y are sideways and Z is upwards. It can rotate horizontally on the world's XY axis (right is clockwise, and left is counter-clockwise), and also ...
2
votes
3answers
52 views

Please help me understand the following notation

Can someone kindly tell me the meaning of the following notation: A book defined the following matrix $(a_{ij})_{3\times 3}$ : $a_{ij}=\begin{cases} d_{ij}& i\neq j\\d_{ii}+\sum_{j=1}^3 d_{ij}&...
0
votes
0answers
11 views

Singular Value Decomposition Basis?

I am unable to just understand one bit of SVD. Given A=USV where U is the eigenvectors of the correlation matrix between the entries S is the eigenvalue matrix and V the transpose of eigenvectors of ...
0
votes
3answers
36 views

Finding the roots of a complex number

I was solving practice problems for my upcoming midterm and however I got stuck with this question type. It is asking me to find all roots and then sketch it. $(1+i\sqrt{3})^{1/2}$ How do we ...
0
votes
1answer
24 views

Calculating the given power of a complex number

$(-2\sqrt{3}+2i)^{-9}$ I tried solving it like I solved other problems, I calculated $r$ which is $4$ then got $4(\sin{\frac{\pi}{6}}+\cos{\frac{\pi}{6}})$, then using an equation, I got $4^{-9}(\...
2
votes
3answers
27 views

When is a nonhomogeneous linear system solvable

I'm supposed to determine for which t the linear system $\bf{A}*x = y$ is solvable $\bf{A}$ = $\begin{bmatrix}1 & 0 & 1\\0 & -1 & 1\\ t-2 & 0 &0 \end{bmatrix},\quad \bf{y} = ...
1
vote
0answers
270 views

Using Cholesky factorization to solve the system $AXA=B$

I have been given a problem of solving $X$, which is an unblurred image, in the system: $$B = A X A \iff X = A^{-1} B A^{-1},$$ where matrix $A$ describes the blurring of an image and matrix $B$ is ...
1
vote
1answer
33 views

Showing a counter-example to the Riesz Representation Theorem in an infinite-dimensional vector space

I'm currently studying Linear Algebra from Sheldon Axler's Linear Algebra Done Right, and there's a problem in Chapter 6.B (when it first introduces orthonormality, the Gram-Schmidt Procedure, and the ...
10
votes
2answers
2k views

On the generators of the Modular Group

The modular group is the group $G$ consisting of all linear fractional transformations $\phi$ of the form $$\phi(z)=\frac{az+b}{cz+d}$$ where $a,b,c,d$ are integers and $ad-bc=1$. I have read that $G$ ...
0
votes
1answer
19 views

Normalizing dual eigenvectors? Why only for trivial defects?

Two little questions to this passage: (1) How can we normalize to get $\langle u,u^*\rangle =1$? (2) Why is this possible if $\lambda$ has trivial defect only (i.e. for trivial Jordan blocks)? I did ...
1
vote
0answers
19 views

Irreducible actions of $\mathbb{Z}^d$ on $\mathbb{T}^n$

I am trying to understand the construction done in this reference http://www.personal.psu.edu/sxk37/pub/KKS-old.pdf by Katok, Katok and Schmidt. In the section 3.3 (p11-13), the idea is to ...
1
vote
1answer
31 views

Eigenvalue and the row sum

How could we prove the following statement? Let $A \in \mathbb C^{n \times n}$ such that $\displaystyle \sum_{j=1}^{n} |a_{ij}| \leq 1$ for all $1 \leq i \leq n$, then $\forall \ \lambda \in \...
1
vote
2answers
47 views

Endomorphism $\phi: M_{n}(\mathbb{C}) \rightarrow M_{n}(\mathbb{C})$ that stabilizes $GL_{n}(\mathbb{C}).$

I've been given this as a homework assignment, and have no idea how to proceed. Can anyone help? The question is: (1) Let $\phi: M_{n}(\mathbb{C}) \rightarrow M_{n}(\mathbb{C})$ be an endomorphism ...
0
votes
0answers
49 views

Finding a solution for $x^2+y^2-z^2 = 1$

Let $$K = \{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 - z^2 = 1 \}$$ Prove that $\mbox{Span}(K) = \mathbb R^3$. How can I solve this and find the vectors that will fit it?
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votes
1answer
28 views

homomorphism question

Check $p(x) + p(-x) ∈ P^e$ for every $p(x) ∈ \mathbb{R}[x]$. Check that the map $Ψ:\mathbb{R}[x]\to P^e$ given by $Ψ(p(x))=(p(x)+p(-x))/2$ is a linear map. Further check that $Ψ^2=Ψ$. Determine ...
1
vote
3answers
50 views

Proving $A^n=\left[\begin{smallmatrix}1&(2^n-1)a\\0&2^n\end{smallmatrix}\right]$ [on hold]

I have this input \begin{bmatrix}1&a\\0&2\end{bmatrix} and I have no clue on how to prove it right. All I could do was to try with $A^2$, $A^3$ and so on but I can't prove it.
0
votes
3answers
22 views

Simultaneous equations result in decimal answer?

I have these 2 equations. $$\left\{ \begin{split} y &= 2x -2 \\ 3y &= -2x + 6 \end{split}\right. $$ I have worked it all out, and plotted the graphs and the point they meet is: $(x,y)=(1,1.5)$....
1
vote
1answer
46 views

fill in table - non-isomorphic groups - permutation.

I am a student in computer science - first year. I study linear linear algebra 2 - course of linear algebra 1. - In some institutions academic studies teach the courses together / teach in another way....
1
vote
0answers
17 views

Linearized operator: Fredholm operator? Jordan form?

Suppose we are given a reaction diffusion equation $$ u_t=u_{xx}+f(u) $$ and search for travelling wave solutions $u(x,t)=U(x-ct)=U(\xi)$ then plugging this into the equation we get $$ U_{\xi\xi}+cU_\...
0
votes
2answers
35 views

If A is a closed linear subspace of a vector space X (which may not be Hilbert) is it true that $A^{\perp} = 0 \Leftrightarrow A = X$?

If A is a closed linear subspace of a vector space X (which may not be Hilbert) is it true that $A^{\perp} = 0 \Leftrightarrow A = X$? $A^{\perp} = \{x \in X|<x,a>=0, \forall a \in A\}$. ...
1
vote
3answers
82 views

How to differentiate the product of vectors and matrices?

Suppose I have $t$, an $m \times n$ matrix of constants and $w$, an $n \times 1$ column vector. I want to differentiate $A$, the function of $w$ defined as $$ A(w) = w^Tt^Ttw. $$ I wish to use ...
0
votes
1answer
33 views

Is it possible to determine the constituents of a matrix product, given the result?

Suppose we have a set $M$ of two or more matrices such that every matrix product $X$ composed of matrices drawn with replacement from $M$ is unique. Is there a set $M$ for which we can determine the ...
0
votes
0answers
431 views

How to differentiate this scalar function written in matrix form, and solve.

Consider $h : \mathbb{R}^+ \to \mathbb{R}$ given as $h(\alpha) = U \{A^2 + 2\alpha AF + \alpha^2 F F^T\}^{-1} U^T$ $U$ is $1\times m$ $A$ is $m\times m$ symmetric PSD. $F$ is $m\times m$ diagonal ...
0
votes
1answer
673 views

Direct sum of kernel and image of two idempotent maps

STATEMENT: Consider two linear maps $q_1,q_2:V \rightarrow V$ such that $q_1\circ q_1=q_1$ and $q_2\circ q_2=q_2$. Assume that $q_1\circ q_2=q_2\circ q_1$. Show that $q_2($ker $q_1)\subseteq$ ker $q_1$...