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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1answer
435 views

Dimensionality of null space when Trace is Zero

This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216). I've succeeded in proving the first three parts, but the most ...
2
votes
2answers
84 views

Prove that $(1, i);(1,-i)$ are characteristic vectors of $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$

Please, help me Prove that $(1, i);(1,-i)$ are characteristic vectors of $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$ I've found the polynomial characteristic: ...
2
votes
0answers
159 views

Cauchy-Binet formula for squares

Using the convention of the wikipedia article, Cauchy-Binet formula states that --for $A, \, n\times m$ and $B, \, m\times n$ matrices-- $$ \det(AB) = \sum_{S\in\tbinom{[n]}m} ...
2
votes
2answers
135 views

linear transformation problem

$T:\mathbb{R}^n\rightarrow \mathbb{R}$ such that $T^2=\lambda T$ for some $\lambda\in\mathbb{R}$ Which of the following are true $||T(x)||=|\lambda| ||x||$ $\forall x\in\mathbb{R}^n$ If ...
2
votes
2answers
431 views

Matrix Multiplication and Function Composition

Given the vector space $F^n$ and two linear function $T,S:F^n \rightarrow F^n$ is it true that multiplying the representative matrices according to the standard basis of $T$ and $S$ is equivalent to ...
0
votes
1answer
120 views

$N$ is a matrix such that $N^3=0$

Given a $3\times 3$ matrix $N$ such that $N^3=0$, then which of the following are/is true? $N$ has a non zero eigenvector $N$ is similar to a diagonal matrix $N$ has $3$ linearly independent ...
1
vote
3answers
253 views

positive semidefinite, positive definite? [duplicate]

Let $a_1,\dots,a_n$ be real numbers, and set $a_{ij} = a_ia_j$. Consider the $n \times n$ matrix $A=(a_{ij})$. Then It is possible to choose $a_1.\dots,a_n$ such that $A$ is non-singular matrix $A$ ...
1
vote
3answers
103 views

A basic question in linear Algebra

Consider the following claim: Let $V$ be a vector space and let $A,B\subseteq V$ be two independent sets with $|A|<|B|<\infty $. Then there exists $b\in B$ such that $A\cup \{b\}$ is ...
4
votes
2answers
291 views

Nearest matrix in doubly stochastic matrix set

Suppose $\mathcal{D}_N$ denote an $N\times N$ doubly stochastic matrix, given any element $M\in \mathcal{D}_N$ , the singular value decomposition for $M$ is $$ M=USV'$$ where $U$ and $V$ are two ...
5
votes
2answers
94 views

How to check that whather a Polygon is completly inside of another Polygon?

Let's say I have two polygons. I know the co-ordinates of both polygons. Now, I need to check whether the first Polygon is completely inside of second polygon? IN this figure only 1 polygon is ...
2
votes
2answers
146 views

Exponential matrix equation

I don't know how to solve (if it's possible) the following matrix equation: $$\exp(H)=H^2,$$ where $H$ is a $N \times N$ hermitian matrix. Does someone know if this equation has solutions and if the ...
2
votes
1answer
200 views

rank for the matrix of concatenating all $N \times N$ permutation matrics

Consider all $N\times N$ permutation matrix $\{M_1,M_2,\ldots,M_{N!}\}$ Define $S_N$ as concatenating each $\operatorname{vec}(M_i)$ as $S_N$'s $i$th column Is there any convenient way to calculate ...
2
votes
4answers
87 views

Derivative of $\|Xa\|_2 $ with respect to $X$

Can someone give me the answer to the following expression? $\frac{\partial}{\partial X}\|Xa\|_2 =?$ $X$ is a square matrix and $a$ is a vektor of the apropriate size. $\|\cdot\|_2$ is the euclidean ...
6
votes
1answer
162 views

$\wedge^k(V)^* \cong \mathrm{Alt}^k(V)$

Let $V$ be a finite dimensional real vector space, let $\mathrm{Alt}^k(V)$ denote the space of alternating $k$-linear forms on $V$ and let $\wedge^k(V)$ denote the $k^{th}$ exterior power of $V$. I ...
1
vote
0answers
51 views

Is the basis vector of a rotated vector in $E^3$ transformed differently than the components of the vector?

Do the basis vectors of a rotated vector in $E^3$ transform differently than the components of the vector? I've recently come across a blog where someone rotated the i,j,k basis vector using the ...
2
votes
5answers
2k views

Are the eigenvectors of a real symmetric matrix always an orthonormal basis without change?

I was reading the wikipedia page for symmetric matrices, and I noticed this part: ...
8
votes
1answer
107 views

Inner product space over $\mathbb{R}$

Definition of the problem I have to prove the following statement: Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$ be an inner product space over $\mathbb{R}$. prove that for all $x,y\in ...
4
votes
2answers
798 views

Matrix commutator question

Here's a nice question I heard on IRC, courtesy of "tmyklebu." Let $A$, $B$, and $C$ be $2\times 2$ complex matrices. Define the commutator $[X,Y]=XY-YX$ for any matrices $X$ and $Y$. Prove ...
1
vote
2answers
579 views

what is derivative of determinant map [duplicate]

Possible Duplicate: Derivative of Determinant Map consider $v=(v_1,v_2)\in \mathbb{R}^2$ ,$w=(w_1,w_2)\in\mathbb{R}^2$ consider the determinant map det:$\mathbb{R}^2\times \mathbb{R}^2$ ...
0
votes
1answer
161 views

linear kernel pca get corresponding dimension

I am implementing my own version of linear kernel principal component analysis for better understanding the algorithm. I faced a problem which seems to be specific ...
1
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1answer
53 views

Small perturbations

Background: Let $x_1,\ldots,x_n$ be the variables satisfying the equations of motion $\ddot{x_i}=f_i(x_1,\ldots,x_n)$ for $i=1,\ldots,n$ We introduce a small perturbation such that $x_i(t)=x_i^0 ...
3
votes
3answers
242 views

Rank and determinant of $D$ , an $n\times n$ real matrix, $n\ge 2$

Let $D$ be a $n\times n$ real matrix, $n\ge 2$. Which of the following is valid? $\det(D)=0\Rightarrow \mathrm{rank}(D)=0$ $\det(D)=1\Rightarrow \mathrm{rank}(D)\neq 1$ $\det(D)=1\Rightarrow ...
3
votes
2answers
499 views

Change of Basis vs. Linear Transformation

If i understand it correctly, change of basis is just a specific case of a linear transformation. Specifically given a vector space $V$ over a field $F$ such that $\dim V=n$, change of basis is just ...
1
vote
2answers
50 views

What is the form of $T$?

Let $T$ is a linear transformation on $\mathbb{R}^2$. $x,y$ linearly indipendent vector in $\mathbb{R}^2$, $T(y)=\alpha x$ and $T(x)=0$, Then with respect to some basis in $\mathbb{R}^2$, $T$ is of ...
1
vote
2answers
123 views

Given a symmetric matrix $A$, are there any matrices $B$, $C$ that $BAC = I$?

Given a $4 \times 4$ symmetric matrix $A$, are there any matrices $B,C$ that: $BAC = I_{4}$ ? I've thought of $B$ being a orthogonal matrix $P$ ($B=P$) and $ C = P^{T}$ so we get $PAP^{T} = ...
1
vote
2answers
352 views

Is there any orthogonal matrix P that makes a symmetric A, diagonal by $PAP^{-1}$?

Given a symmetric matrix A. Is there any orthogonal matrix P that makes $PAP^{-1}$ diagonal? I've found at wikipedia this: The finite-dimensional spectral theorem says that any symmetric matrix ...
5
votes
2answers
7k views

In which cases is the inverse matrix equal to the transpose?

As said in the title, in which cases an invertible matrix is equal to the transpose? When is this: $ A^{-1} = A^{T} $ true? If the matrix A is orthogonal? Thank you!
2
votes
1answer
286 views

How to make derivative operation in matrix space?

\begin{equation}\frac{d}{d\theta}\frac{1}{2}(\theta^TX - y)^2 = 0\end{equation} where, $X$ is $m $ on $ n$ matrix, $y$ is $m$-dimensional vector, $\theta$ is n-dimensional vector. I can solve this ...
1
vote
1answer
208 views

How to prove linear independence using linear functionals in dual space?

I'm reading deBoor's (wonderful) book "A practical guide to splines", revised edition. I'm doing some of the exercises at the end of each chapter just to fix the main ideas before going ahead... ...
2
votes
2answers
68 views

Find $A \in M_{2}(\mathbb{Z})$ such that $M_{2}(\mathbb {Z})=\{\sum a_{i}A^{i} : a_{i} \in \mathbb{Z}\}$

Question: Does there exist $A \in M_{2}(\mathbb{Z})$ such that every element of $M_{2}(\mathbb{Z})$ can be represented as a linear combination of powers of $A$ with integer coefficients? In other ...
0
votes
2answers
53 views

Is there a simple way of arriving at this solution?

Suppose we are given the matrix $$\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}\cos(\omega t)& -\sin(\omega t)\\\sin(\omega t)& \cos(\omega ...
2
votes
0answers
117 views

Solution for this matrix equations (closed form or approximate solution)

Given a system of equations, I'm curious whether I can find the closed form solution for $P$, Here, $G$,$H$ are known $N \times N$ matrix, $\lambda$ is a known scalar; $s$,$t$ are two $N \times 1$ ...
1
vote
1answer
59 views

operators polynomials

Let $T:V \longrightarrow V$ a linear operator, where $V$ is a vector space over the field $\mathbb{K}$. Show that if $p(x),q(x)\in \mathcal{P}(\mathbb{K})$, then $$(p\cdot q)(T)(v)=p(q(T))(v), \ \ ...
1
vote
2answers
89 views

Spanning a vector with no zero coordinates

Given a complex square matrix with 1-s on the main diagonal (and arbitrary values elsewhere), do its columns span a vector with no zero coordinate? Clarification: What I'm asking is, given a complex ...
3
votes
1answer
190 views

Tensor products of maps

Let $V, W, U, X$ be $R$-modules where R is a ring. At what level of generality, if any is it true that the maps (I always mean linear) from $V \otimes W$ to $U \otimes X$ can be identified with $L(V, ...
5
votes
2answers
310 views

Special orthogonal matrix uniquely determined by $n-1 \times n-1$ entries?

For example, consider the specific question: Given $a_{11},a_{12},a_{21},a_{22}$ does that uniquely determine $A=\begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ ...
2
votes
1answer
252 views

Reference for linear independence of a set of pairwise-independent irrationals

I've found several uses of the phrase "linearly independent over the rationals" that imply that any set of irrationals is linearly independent over the rationals if it is pairwise linearly independent ...
6
votes
2answers
272 views

Determinant called Grammian

Famously, if functions $f_1,f_2,…,f_n$, each of which possesses a derivative of order $n-1$, are linearly independent on the interval $I$, if $$ \det\left( \begin{array}{ccccc} f_1 & f_2 & ...
13
votes
3answers
459 views

Does there exist a 3-dimensional subspace of real functions consisting only of monotone functions?

This is Exercise 1.O from the book Van Rooij, Schikhof: A Second Course on Real Functions. The set of the monotone functions on $[0,1]$ contains all polynomial functions of degree $\le 1$. ...
1
vote
0answers
63 views

Looking for a reflection of 30° at a line

I'm trying to find a matrix-expression of a 30° reflection at the line $g(x)=2x+4$ Somebody can give a hint? Greetings
0
votes
3answers
1k views

Real life application of Gaussian Elimination

I would normally use Gaussian Elimination to solve a linear system. If we have more unknowns than equations we end up with an infinite number of solutions. Are there any real life applications of ...
2
votes
2answers
2k views

How to get Point between two points at any specific distance?

I have two points, approximately we take values for that: Point $A = (50, 150)$; Point $B = (150, 50)$; So the distance should be calculated here, $\text{distance} = \sqrt{(B_x - A_x)(B_x - A_x) + ...
4
votes
1answer
148 views

Block Determinants

This is a nice question I recently found in Golan's book. Problem: Let $A,B,C,D$ be $n\times n$ matrices over $\mathbb{R}$ with $n\ge 2$, and let $M$ be the $2n\times 2n$ matrix \begin{bmatrix} A ...
56
votes
3answers
2k views

Cute Determinant Question

I stumbled across the following problem and found it cute. Problem: We are given that $19$ divides $23028$, $31882$, $86469$, $6327$, and $61902$. Show that $19$ divides the following determinant: ...
7
votes
1answer
216 views

Existence of the Pfaffian?

Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes. If $n$ is even, my book claims that the ...
4
votes
4answers
1k views

Help Understanding Proof of Replacement Theorem?

Sorry if this is a trivial question. The book is Linear Algebra Done Right by Axler, page 25-26. Theorem: In a finite-dimensional vector space, the length of every linearly independent list of ...
4
votes
3answers
816 views

Partial derivative of trace of an inverse matrix

I have the following vector function $f(\mathbf{x})=\operatorname{Tr}[(\mathbf{A}+\operatorname{diag}(\mathbf{x}))^{-1}]$ where $\operatorname{diag}(\mathbf{x})$ is the diagonal matrix with values ...
0
votes
1answer
571 views

Purpose of Inverse matrix

What use is the inverse matrix? I would not use it to solve linear systems but there must be some concrete or real life applications where it is used.
1
vote
1answer
349 views

Closed-form for eigenvectors of rotation matrix

For matrices that are elements of $SO(3)$ is there a formula for the eigenvectors corresponding to the eigenvalue $1$ in terms of the entries of the matrix?
5
votes
1answer
558 views

Eigenvalues of $A+B$

$A,B$ are symmetric matrices, $A$ has eigenvalues in $[a,b]$ and $B$ has eigenvalues in $[c,d]$ then we need to show that eigenvalues of $A+B$ lie in $[a+c,b+d]$, I am really not getting where to ...