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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459 views

Finding the transformation when given transformation matrix

Lets say, there is a transformation: $T:\Re ^{n}\rightarrow \Re ^{m}$ transforming a vector in $V$ to $W$. Now the transformation matrix, $A=\begin{bmatrix} a_{11} & a_{12} &...&a_{1n} \\ ...
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3answers
205 views

Isomorphism from $Hom(V,W)$ to $M_{m\times n}^F$

Our book states that there is an isomorphism from $Hom(V,W)$, the vector space of all linear transformations from $V$ to $W$, to the matrix space $M_{m\times n}^F$ which is defined by $T:\rightarrow ...
2
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2answers
96 views

Linear algebra endomorphism exercise

This problem is taken from Golan's linear algebra book. Problem: Let $V$ be an inner product space and $\alpha$ an endomorphism satisfying $\alpha^*\alpha=0$. Show $\alpha=0$.
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1answer
70 views

Easy condition for positive definite endomorphism

This problem is taken from Golan's linear algebra book. Problem: Let $V$ be an inner product space over $\mathbb{R}$ and let $\alpha$ be an endomorphism of $V$. Show that $\alpha$ is positive ...
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2answers
77 views

If an endomorphism satisfies $\alpha^* = -\alpha$, then its eigenvalues are purely imaginary

This is another exercise from Golan's book. Problem: Let $V$ be an inner product space over $\mathbb{C}$ and let $\alpha$ be an endomorphism of $V$ satisfying $\alpha^*=-\alpha$, where $\alpha^*$ ...
2
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1answer
406 views

Floor tile problem

The foor of a rectangular room is covered with equal numbers of red and blue square tiles. The room is x tiles wide and y tiles long. If only red tiles are placed around the edge of the room and all ...
2
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2answers
122 views

In which case $M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$ is true?

Usually for modules $M_1,M_2,N$ $$M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$$ is wrong. I'm just curious, but are there any cases or additional conditions where it gets true? James B. ...
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2answers
124 views

A question on symmetric matrix and application of Spectral theorem.

Today in the class Prof. applied spectral theorem and wrote $A$ a semidefinite positive matrix as $A=\sum \lambda_i v_i\times v_i$ , where $v_i$ are the eigenvectores and $\lambda_i$ are ...
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1answer
372 views

Linear programming: the optimum of the shortest path problem is attained by $x \in [0, 1]^m$

Let $G=(V,E)$ be a graph, where $|E|=m$, and suppose we formulate the shortest path problem on $G$ as follows: minimize ${}^t(1,\dots,1)x$ such that $Bx={}^t(1,-1,0,\dots,0), x\in \{0,1\}^m$, where $B ...
3
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3answers
872 views

Linear algebra: orthogonal projection?

(a) Find the orthogonal projection of $(-1, 0, 8)$ onto the normal vector to the plane $x-2y+z=0$. Is this question saying to find the orthogonal projection in other words? The way the question is ...
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2answers
545 views

Show trace is zero

Problem: We are given $n\times n$ square matrices $A$ and $B$ with $AB+BA=0$ and $A^2+B^2=I$. Show $tr(A)=tr(B)=0$. Thoughts: We have $tr(BA)=tr(AB)=-tr(BA)=0$. We also have the factorizations ...
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1answer
239 views

Finite set of matrices closed under multiplication

The following problem is from Golan's linear algebra book. I have been unable to make headway. Problem: Let $n\in\mathbb{N}$ and $U$ be a non-empty finite subset of the $n\times n$ matrices over ...
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1answer
127 views

Linear dependence of linear functionals

Problem: Let V be a vector space over a field F and let $\alpha$ and $\beta$ be linear functionals on $V$. If $\ker(\beta)\subset\ker(\alpha)$, show $\alpha = k\beta$, for some $k\in F$. A proposed ...
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1answer
86 views

Minimal polynomial over the complex field

Is there a matrix with real entries such that its minimal polynomial with coefficients in the complex field is different than its minimal polynomial over the real numbers?
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1answer
281 views

Matrix with no eigenvalues

Here is another problem from Golan. Problem: Let $F$ be a finite field. Show there exists a symmetric $2\times 2$ matrix over $F$ with no eigenvalues in $F$.
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1answer
126 views

Is there any closed-form expression to calculate each element of the inverse of a matrix?

Considering a generic square matrix $A=(a_{i,j})$ we want to compute its inverse $A^{-1}=\left[a^{(-1)}_{i,j}\right]$. Is there a way to express each $a^{(-1)}_{i,j}$ using a closed form expression?
5
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1answer
181 views

Complex matrices with null trace

I'm trying to prove the following: Let $A\in \mathbb{C}^{n\times n}$ be a matrix with null trace; then $A$ is similar to a matrix $B$ such that $B_{jj}=0$ (i.e. it has zeroes on its diagonal). ...
8
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1answer
426 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
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2answers
83 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: ...
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0answers
148 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
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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
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2answers
420 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 ...
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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 ...
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3answers
251 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 $A$ ...
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3answers
101 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
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2answers
286 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
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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
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1answer
190 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
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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
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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 ...
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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
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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
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1answer
106 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
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2answers
777 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 ...
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2answers
556 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
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1answer
160 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 ...
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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 ...
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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
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2answers
488 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 ...
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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 ...
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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} = ...
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2answers
348 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
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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
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1answer
278 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 ...
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1answer
205 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... ...
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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 ...
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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
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0answers
113 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 $N \times 1$ ...
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1answer
58 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), \ \ ...