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 ...

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If $N$ is nilpotent of index $n\geq 2$ but $N^{n-1}\neq 0$ then there's no $A$ such that $A^2=N$

Let $N\in M_{n\times n}^{\mathbb{C}}$ a nilpotent matrix of index $n\geq 2$. Prove: if $N^{n-1}\neq 0$ then there does not exist a matrix $A\in M_{n\times n}^{\mathbb{C}}$ such that $A^2=N$. My ...
2
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1answer
22 views

Matrix with all 1's diagonalizable or not?

This is a followup to my question here. Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. Is $A$ diagonalizable?
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0answers
4 views

Under what condition given $(x_1, y_1\cdot r_1),…,(x_n, y_n\cdot r_n)$ we can interpolate polynomial $T$ that has specific random root?

We know given $(x_1, y_1),...,(x_n, y_n)$ we can interpolate a polynomial $P$ of of degree at most $n-1$. Let us define polynomial $P=(x-\beta)\cdot g(x)$, where degree of $P$ is at most $n-1$, ...
1
vote
0answers
5 views

dimension of intersection of subspaces

In a 13-dimensional vector space, the dimension of intersection of two 6-dimensional subspaces is 1. at least 1 2. at most 1 3. at least 6 4. at most 6 My thought:For two subspaces U and W of vector ...
4
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0answers
10 views

$GL_n(\mathbb{C})$-conjugation invariant $\mathbb{C}$-valued polynomial has certain form.

What is the easiest way to see that any $GL_n(\mathbb{C})$-conjugation invariant $\mathbb{C}$-valued polynomial function on $M_n(\mathbb{C})$ has the form $f(A) = F(p_0(A), \dots, p_{n-1}(A))$ for ...
4
votes
2answers
32 views

Are vectors $e_i$ linearly independent if and only if the matrix $A$ is nonsingular?

Let $V$ be an inner product space over $\mathbb{R}$ or $\mathbb{C}$, and let $e_1, e_2, \dots, e_n$ be a collection of vectors in $V$, not necessarily orthonormal. (Here, $n$ has nothing to do with ...
2
votes
3answers
41 views

Eigenvalues of matrix with all $1$'s.

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
0
votes
1answer
9 views

Proof for the fact that the method of characteristic equations is a valid method to solve recurrence relations

Could someone kindly point me to a proof of the fact that the method is characteristic equations is a valid way of solving recurrence relations? It seems fairly arbitrary to me. I would be grateful ...
0
votes
0answers
9 views

Project point on plane - Unique identfier?

I have a number of planes (in $\mathbb{R}^3$), each represented by a point $\vec{P_i}$ which lies within each plane and the normal vector $\vec{n_i}$. If I project a point $\vec{Q}$ (which does not ...
0
votes
1answer
22 views

Is a matrix similar to its RREF?

Let a matrix be denoted by A and its RREF be denoted by R. Then, is it true that R is similar to A? I am trying to find out Jordan canonical form of a large matrix. If I can somehow prove that a ...
5
votes
2answers
30 views

If $N$ is nilpotent then there exists $A$ such that $A^2=I+N$

Suppose $N\in M_{3\times 3}^{\mathbb{C}}$ is a nilpotent matrix. Prove that there exists $A\in M_{3\times 3}^{\mathbb{C}}$ such that $A^2=I+N$. Hint: find $A$ in the form $A=P(N)$ where $P$ is a ...
0
votes
1answer
13 views

Is Bs(1,-1) linear?

I would like to prove that the Baumslag-Solitar group $BS(1,-1)=\langle a,b| bab^{-1}=a^{-1}\rangle$ is embeddable in $GL_n(\mathbb{Z})$ for some nonnegative integer $n.$ So i tried to find two ...
0
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0answers
9 views

about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$

I am a little confused about the complex conjugation of a complex vector space $V\left(\mathfrak{\mathcal{C}}\right)$. From other answers (Is a complex vector space closed under complex ...
0
votes
1answer
28 views

Minimal Polynomial

I have found the following characteristic polynomial: $$(x+2)(x-2)^2$$ I need to write down all the possible minimal polynomial, so I wrote: $${(x+2),(x-2),(x+2)(x-2),(x+2)(x-2)^2,(x-2)^2}$$ Why is ...
1
vote
0answers
21 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
2
votes
1answer
46 views

$A^k = I$ implies diagonalizable? [duplicate]

If $A$ is a square complex matrix with $A^k = I$ (where $I$ is the identity matrix of the same size as $A$) for some positive integer $k$, does it follow that $A$ is diagonalizable?
5
votes
3answers
38 views

$\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
0
votes
1answer
18 views

Prove $2a^Tb \leq \|a \|^2 + \|b\|_*^2$ with dual norm

We know we can easily (**) prove $$ 2a^Tb \leq \|a \|_2^2 + \|b\|_2^2, \forall a,b $$ Is there a way to prove the following: $$ 2a^Tb \leq \|a \|^2 + \|b\|_*^2, \forall a,b $$ where ...
0
votes
0answers
11 views

norm on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
0
votes
2answers
33 views

Finding eigenvalues and eigenvectors of $2 \times 2$ matix

I having a few issues finding the eigenvectors for the following matrix: $$ \begin{bmatrix} -1 & -1\\ 0 & -2 \\ \end{bmatrix}$$ I calculated the eigenvalues to be ...
4
votes
1answer
22 views

Exist basis, simultaneously upper-triangular?

Let $A, B \in M_n(\mathbb{C})$ be such that $\text{rank}(AB - BA) \le 1$. Does there exist a basis of $\mathbb{C}^n$ with respect to which $A$ and $B$ are simultaneously upper-triangular?
2
votes
1answer
16 views

For a linear system, why is direction “stored” in the variables when considering it as linear equations, but in vectors when its as a vector equation?

Given an arbitrary system of equations, why is direction in space "stored" in the variables when considering the system as linear equations, but "stored" in vectors when considering the system as a ...
-5
votes
1answer
28 views

Is the free abelian group of rank 2 linear?

Is the group $\mathbb{Z}^2$ linear? By linear I mean There is a injective homomorphism from $\mathbb{Z}^2$ to $GL_n(\mathbb{Z})$ for some nonnegative interger $n.$ I tried the following homomorphism ...
0
votes
2answers
30 views

$f$ is a differentiable map and compute $Df(A)(H)$.

Let $f : GL(n, \Bbb R) \to GL(n, \Bbb R)$ be defined by $f(A) = A^{-1}$ where derivative of the matrix $A$ exists. Then $f$ is a differentiable map and compute $Df(A)(H)$. $A A^{-1} = I \implies ...
0
votes
1answer
23 views

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$? If this is false in general, is it possibly true for nilpotent ...
1
vote
1answer
28 views

Peculiar family of apparently positive semidefinite matrices

Let $x_1, \dots, x_n > 0$ be positive real numbers. From numerical experiments, it appears that the $n \times n$ matrix $$A_{ij} = \frac{1}{x_i + x_j} $$ is always positive semidefinite. Is ...
2
votes
3answers
39 views

Basis for subspace in $\mathbb{R}^4$

How would I start to answer this: Show that the vectors $(1,0,0,1)$, $(0,1,0,1)$, and $(0,0,1,1)$ form a basis for the subspace $V$ of $\mathbb{R}^4$ which is defined by the equation ...
0
votes
1answer
23 views

$m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

As the title suggests, how do I see that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
1
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0answers
35 views

About a matrix identity.

In a document named as "The Matrix Cook-Book" I saw two expressions of which I do not get any clue how they are derived. For $n = 3:$ $\det(I + A) = 1 + \det(A) + Tr(A) + 1/2\ Tr(A)^2 − 1/2\ ...
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votes
1answer
31 views

linear algebra (norm) [on hold]

Can someone explain to me the following definition - $\|T\|$ := $ \sup \{\|T(v)\| : v \in \mathbb{R}^n, \|v\| = 1\}$ where $T$ is a linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ and ...
-1
votes
0answers
25 views

How do you find the null space of an inconsistent system? [on hold]

For example, the augmented matrix: $$\left(\begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) $$
0
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0answers
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quadratic form associated with projection operator in Hilbert space

we are in Hilbert space $L^2 $ 1) we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
5
votes
1answer
27 views

Finding an explicit eigenvector

Let $A$ be an $n\times n$ matrix over a field and let $\operatorname{adj}(A)$ denote its classical adjoint. Suppose all column sums of $A$ are zero so that $A$ is singular. If $\operatorname{rank}(A) ...
0
votes
0answers
11 views

Geometrical interpretation of the condition number as measure of matrix dissimilarity

Consider two $p$ by $p$ symmetric positive definite matrices $\pmb F$ and $\pmb G$ and denote $$\pmb D=\pmb G^{-1/2}\pmb F \pmb G^{-1/2}.$$ Sometimes, the condition number of $\pmb D$ will be used ...
1
vote
2answers
47 views

Distinct eigenvalues and matrices problem

Let $V$ be a real vector space and $T: V \rightarrow V$ be a linear transformation. It is given that if $v_1, . . . , v_n$ are eigenvectors for distinct eigenvalues $λ_1, . . . λ_n$ then $\{v_1, . . ...
1
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0answers
27 views

A question about minimizing the $\lambda_{max}$ over a set of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ entry matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (if it helps you can assume that $(1)$ all the ...
4
votes
1answer
39 views

Simultaneous orthogonal diagonalization of two matrices

Let $A=\begin{pmatrix} 1 & -2\\ -2 & 5 \end{pmatrix}$ and $B=\begin{pmatrix} -3 & 6\\ 6 & -10 \end{pmatrix}$. Obviously $A$ is positive-definite and thus we can simultaneously ...
1
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0answers
13 views

Action of the Symplectic Group on Siegel Upper Half Plane

Given $G= \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in Sp_{2n}(\mathbb{R})$ one can define an action on the symmetric $n \times n$ complex matrices with positive definite imaginary part by ...
4
votes
5answers
86 views

Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$

It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$ It is equivalent to the first order system in two variables $$\begin{bmatrix} x' \\ y' \end{bmatrix} = ...
0
votes
0answers
47 views

Good algebra book to cover these topics?

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your ...
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0answers
30 views

The Intution Behind Real Symmetric Matrices and Their Real Eigenvectors

I am wondering about the geometric intuition behind real symmetric matrices and their corresponding linear transformations. Is it possible to understand geometrically why real symmetric matrices ...
1
vote
1answer
20 views

Finding the minimal polynomial of a linear operator

Let $P=\begin{pmatrix} i & 2\\ -1 & -i \end{pmatrix}$ and $T_P\colon M_{2\times 2}^{\mathbb{C}} \to M_{2\times 2}^{\mathbb{C}}$ a linear map defined by $T_P(X)=P^{-1}XP$. I need to find the ...
1
vote
1answer
65 views

Matrix exponential of $\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$

It is asked to evaluate the matrix exponential of $$A=\begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix}$$ It is not hard to do this, since this matrix have 3 ...
0
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0answers
32 views

Row rank$=$Column rank

This is one of the proofs given on Wikipedia. Let $A$ be an $m \times n$ matrix with entries in the real numbers whose row rank is $r$. Therefore, the dimension of the row space of $A$ is $r$. Let ...
0
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0answers
20 views

Distance/Similarity between matrices (different size) [on hold]

I have many matrices that have different size. Specifically, those matrices have the same number of rows but vary in the number of column. Each row is a different signal measurements, and each column ...
0
votes
1answer
11 views

Verifying expression with MP Pseudoinverse

Numerical simulations suggest that the expression $$ A=G^\dagger (PGG^\dagger P)^+G, $$ where $^+$ denotes the Moore-Pensore pseudoinverse, $P$ is the projector $$ P=I-\frac{1}{c^\dagger G^\dagger G ...
0
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0answers
27 views

Kernel of homomorphism

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I wanna show that the following homomorphisms $f_1$ and $f_2$ defined by $f_1: H\to GL_n(\mathbb{Z})$ $f_1(p)=P$ and $f_1(q)=Q$ ...
1
vote
1answer
29 views

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$.

Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$. I'm not sure how to find these sets. I'm sure there is an elementary solution. Any solutions ...
0
votes
2answers
23 views

Row sum of $P^{m}$ when row sum of $P$ is $1$

Let $P$ be an $n\times n$ matrix whose row sum equals $1$. Then for any positive integer $m$ , what is the row sum of $P^{m}$ ? Now I took arbitrary $2\times 2$ matrix ...
0
votes
0answers
14 views

Connected components of pseudospectra

In this Article, page 5 Theorem 2.3 ,what is connected components of pseudospectra of matrix polynomial?