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 ...

learn more… | top users | synonyms

2
votes
1answer
106 views

AX = b has solution iff $\langle A, Y\rangle = 0$ implies $\langle b,Y\rangle = 0$

Let $A$ be an $m\times n$ matrix of real numbers, $b \in \mathbb{R}^n$. Show that the system of equations $AX = b$ has a solution if and only if $A^TY = 0$ always implies $b^TY = 0$. I'm not sure ...
4
votes
0answers
184 views

Invariant of matrix under elementary transformations

$\DeclareMathOperator{\rank}{rank}$ Let $A \in \mathbb R^{n \times n}$, $b \in \mathbb R^n$, $c \in \mathbb R$. Consider the following matrix $$ B = \begin{bmatrix} A & b \\ b^T & c ...
0
votes
2answers
41 views

$T^2$ and $N(T^2)$: What are they?

What is $T^2$ and what does $N(T^2)$ mean? Any help is appreciated. The transformation is applied twice? Is that it?
0
votes
1answer
78 views

Linear Algebra - Linear transformation

Let $T: \mathbb R^2 \rightarrow \mathbb R^2$ be a linear transformation. Does there exist nonzero vectors $u, v, w$ such that $$ T(u)+u= T(v)+2v=T(w)+3w=0\ ? $$
1
vote
2answers
120 views

Proving that matrix $A$ is a scalar matrix if $C^{-1} A C$ is diagonal

I've been stuck with this question: Given $A \in M_{n\times n} (\mathbb F)$ and $\forall C \in M_{n\times n}(\mathbb F) , C \; \text{is invertible} \; \rightarrow \; C^{-1}AC \; \text{is ...
1
vote
1answer
106 views

A vector space V is an irreducible End(V)-module

Let $V$ be a nonzero vector space. I consider as a $\operatorname{End}(V)$-module. Then it is irreducible. My thoughts: Let $U$ be a nonzero submodule and $u\in U-\{0\}$. I want to show that $U=V$. ...
4
votes
5answers
438 views

Is $B = A^2 + A - 6I$ invertible when $A^2 + 2A = 3I$?

Given: $$A \in M_{nxn} (\mathbb C), \; A \neq \lambda I, \; A^2 + 2A = 3I$$ Now we define: $$B = A^2 + A - 6I$$ The question: Is $B$ inversable? Now, what I did is this: $A^2 + 2A = 3I ...
0
votes
0answers
63 views

How prove Matrices A and B inertia index are equality

Let $A,B$ be Hermitian matrices,and the real eigenvalues such that:$Re{(\lambda(AB))}\ge 0$. Show that Matrices A and B inertia index are equality.
3
votes
1answer
104 views

Pseudo inverse interpretation

Consider two integers $m$ and $n$, with $m > n$, and $A$, $x$ and $b$ real matrices and vectors. In the case $A x = b$, with $A$ of dimension $m \times n$ (and therefore $x$ of dimension $n \times ...
1
vote
1answer
180 views

Cancellation property in matrices.

I just found a question which is based on a doubt I have carrying for over 10 years. If $ACC^t=BCC^t$ : $C^t$ means transpose of $C$ Is $A=B$ $AC=BC$ Sorry if this is a trivial question. ...
7
votes
2answers
171 views

Prove that if $A$ is diagonalizable then there is a matrix $B$ such that $B^{2012} = A$

Given: $$A \in M_{n\times n} (\mathbb C) \; , \; A \; \text{is diagonalizable}$$ We need to prove that: $$ \exists B \in M_{n\times n} (\mathbb C) \; : B^{2012} = A$$ What I said so far: If $A$ ...
0
votes
1answer
92 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
1
vote
0answers
24 views

How do i prove that "every monic polynomial is the characteristic polynomial of some matrix? [duplicate]

Let $F$ be a field and $f(X)\in F[X]$ be a monic polynomial such that $deg(f(X))=n$. How do i prove that there exists $A\in\mathscr{M}_n(F)$ such that $\det(XI_n - A)=f(X)$? I checked the solution ...
2
votes
2answers
309 views

How many row operations must for A and B to be row equivalent?

This definition is from Hoffmann and Kunze. Definition. If $A$ and $B$ are $m \times n$ matrices over the field $F$, we say that $B$ is row-equivalent to $A$ if $B$ can be obtained from $A$ by ...
0
votes
1answer
39 views

$N(T^n) = N(T^{n+1})$

Let $T : V \rightarrow V$ be a linear map and $\dim(V ) < \infty$. How can I show that there is an $n > 0$ such that $N(T^n) = N(T^{n+1})$? Could someone here possibly help?
1
vote
1answer
76 views

Prove that in a vector space $V$ over field $\mathbb{F}$ $0\cdot v=0$

Prove that in a vector space $V$ over field $\mathbb{F}\space$: $0\cdot v=0$ for all $v \in V$ I started by proving that for all $x \in F$ $x \cdot 0 = 0$ using the Field axioms. Then I said that ...
1
vote
1answer
47 views

Does the Border (Boundary) Points of a convex shape in the positive quadrant make a convex function?

Let $\mathbb{S}$ be a convex body in 2-D with some non-zero intersection with positive quadrant and let it also contain origin. Let $c>0$ be the right-most point on the x-axis such that $(c,y)\in ...
0
votes
1answer
97 views

Prove that $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is positive definite

How to prove if A is a positive definite matrix, then $(X'X)^{-1}X'AX(X'X)^{-1}-(X'A^{-1}X)^{-1}$ is also positive definite? Here $X'$ denotes the transpose of $X$. $A$ is square and $X$ is $n\times ...
2
votes
1answer
84 views

On triangular decomposition of square matrix

Let $L\in Gl_n(\mathbb{C})$ and define $A=LL^*$. Let us consider another decomposition such as $A=L_1L_1^*$. What is the relation between $L$ and $L_1$. One obvious relation is $L_1=LU$ where $U$ is ...
0
votes
2answers
609 views

Let a, b, c be three nonzero vectors, any two of which are perpendicular. Prove that these 3 vectors are linearly independent.

Here is my answer: We assume two cases, and work to prove that both assumptions are incorrect, leading to a proof. Without loss of generality, assume that $ \mathbf{a} $ is linearly dependent with ...
2
votes
4answers
494 views

$ax+by+cz=d$ is the equation of a plane in space. Show that $d'$ is the distance from the plane to the origin.

This is a 3 part practice question I would like to get some feedback on. I think I have solved the 1st two parts, but I need a little direction for part (c) (the title is Part (a) ) which is repeated ...
2
votes
2answers
332 views

The relationship between eigenvalues of matrices $XY$ and $YX$

If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove that $\lambda (XY) = \lambda (YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$? Here, ...
0
votes
1answer
28 views

Expressing a transformation matrix

Let $B=\{v_1,...,v_n\}$ and $C=\{w_1,...,w_n\}$ be bases to $V$. Suppose: $w_i=m_{i1}v_1+...+m_{in}v_n$ for $m_{ij}\in F, 1\le i,j \le n$. $M$ is an invertible matrix whose ($i,j$) member is $m_{ij}$. ...
4
votes
1answer
84 views

Optimal rotation to align a circle with external points

I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
3
votes
1answer
80 views

Diagram Chase Argument: $\text{rank}(T)=\text{rank}([T]^{\scr{C}}_{\scr{B}})$

\begin{eqnarray} \text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \end{eqnarray} Theorem Let $T:V\rightarrow W$ be a linear map with $\scr{B}$ and $\scr{C}$ bases of $V$ ...
2
votes
2answers
63 views

Is the linear dependence test also valid for matrices?

I have the set of matrices $ \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix} $ $ \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ ...
1
vote
1answer
608 views

Underdetermined linear systems least squares

I have an underdetermined linear system, with 3 equations and four unknows. I also know an initial guess for these 4 unknows. The article I am reading says: We can solve the system using the least ...
1
vote
1answer
251 views

How would I find this eigenvalue?

I'm told to let $A$ be the matrix of the linear transformation $T$ and without writing $A$, find an eigenvalue of $A$ and describe the eigenspace. The first is to let $T$ be the transformation on ...
2
votes
1answer
571 views

Limit of matrix powers.

Consider an arbitrary matrix $A$ with eigenvalues within the unit circle. Is there a nice formula for $A^\infty = \lim_{n \rightarrow \infty} A^n$? In particular, maybe there is a formula which ...
4
votes
1answer
156 views

Angular alignment of points on two concentric circles

I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
3
votes
5answers
382 views

Prove that if $A - A^2 = I$ then $A$ has no real eigenvalues

Given: $$ A \in M_{n\times n}(\mathbb R) \; , \; A - A^2 = I $$ Then we have to prove that $A$ does not have real eigenvalues. How do we prove such a thing?
1
vote
1answer
49 views

Angle consistency between vectors in N dimensions

I am trying to understand how rotations work in higher dimensions. Let us assume we have a set of points $p_i\in P$ in $N$ dimensions, related to another set of points $q_i \in Q$ by a rotation $R$. ...
2
votes
1answer
2k views

Show AB and BA have the same eigenvalues [duplicate]

If $A$ and $B$ are $n$ by $n$ matrices show that $AB$ and $BA$ have the same eigenvalues. I see why this is true if both are nonsingular. But does it still hold if they are not invertible? Thanks!
2
votes
2answers
113 views

Finding for every parameter $\lambda$ if matrix is diagonalizable

Given: $$A = \begin{pmatrix} 1 & i & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & i \end{pmatrix} \; , \; \lambda \in \mathbb C$$ For every value of $\lambda$ I have to know if the matrix ...
0
votes
4answers
92 views

Prove that if $A \in M_{2\times2}\mathbb {(R)}$ is symmetric then A is diagonalizable

Given that: $$A \in M_{2\times2} \mathbb {(R)}$$ we have to prove that $A$ is diagonalizable. As in: $$\text{There exists a turnable matrix } P \; (\text{det(P) != 0 }) \; \text{such that}:$$ ...
0
votes
2answers
48 views

Quick question about proofs of theorem concerning Jordan basis

I have a question about proofs of this theorem: Let $K$ be an algebraically closed field, $V$ be a finite-dimensional space over $K$ and $f : V → V$ be a linear operator. Then there exists a Jordan ...
0
votes
1answer
37 views

Inverse of Eigen value

What is the physical meaning of inverse square root of the eigen value? Is it possible to use it as stretch factor to decorrelate the data.
2
votes
2answers
139 views

Minimum distance of the linear code $\{0,1\}$

Let $H$ be a check matrix for a linear code $C$. Then the minimum distance of $C$ is $d \in \mathbb N$ such that there exists a set of $d$, but no set of $d-1$, linearly dependent columns in $H$. ...
10
votes
1answer
188 views

Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code

I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection. Let $C$ be some extended ...
1
vote
0answers
50 views

Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
1
vote
0answers
222 views

Showing no non-trivial t-invariant subspace has a t-invariant complement.

The question is from Hoffman and Kunze Let T be a linear operator on a finite-dimensional vector space V. Suppose that: (a) the minimal polynomial for T is a power of an irreducible ...
5
votes
3answers
70 views

$f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=…$

I am stuck on the following problem: Let $f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=(x_2+x_3,x_3+x_1,x_1+x_2).$ Then the first derivative of $f$ is : 1.not invertible ...
1
vote
1answer
726 views

Linear Algebra: Least-Squares Approximation & “Normal Equation”

I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
2
votes
1answer
67 views

I need to diagonalize this matrix but I'm not sure it can be

This is the matrix I need to diagonalize: $A=\left[\begin{matrix}3&2\\0&3\end{matrix}\right]$. So I found the eigenvalue by taking the determinant of $(A-\lambda I)$ and solving for ...
1
vote
1answer
130 views

Find all eigenvalues and corresponding eigenvectors for the matrix?

Find all eigenvalues and corresponding eigenvectors for the matrix: $$ \left(\begin{array}{cr} 0&-1 \\ 2&3 \end{array}\right) $$ Not looking for a answer, but I don't know what an "eigenvalue" ...
0
votes
3answers
154 views

Underdetermined System and Minimizing Cost

I need to minimize 4x + 4y subject to the following constraints: $4x + 8y = 40$ $x + 2y = 10$ Any ideas? Answers must be integers, as they represent physical units.
4
votes
1answer
83 views

Rank after addition of positive definite matrices

I have two positive semidefinite matrices $A$ and $B$. Is it necessarily true that $$ rank(A+B) = rank(A^2+A+B) $$ ? It is easy to see that $rank(A+B) \le rank(A^2+A+B)$, but for any example I try, ...
2
votes
2answers
139 views

Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
1
vote
1answer
153 views

how to calculate the “variance OF the covariance” matrix : E[vech(x x') vech(x x')'] for normal distributed x?

Supposing a vector x follows normal distribution. I want to calculate the expectation of the "variance Of the covariance matrix" (not variance-covariance matrix) in a vector form, meaning E[vech(x ...
0
votes
2answers
53 views

Matrix Algebra (Elementary)

I have $\hat\xi =\lambda_1\textbf{1V}^{-1} + \lambda_2\textbf{rV}^{-1}$ and sub it in to my two constraints, namely, $\xi\textbf{1}^T = 1$ and $\xi\textbf{r}^T = \mu$. My lecture notes then say set ...