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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2
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2answers
170 views

how to find all the solutions to $I+A+\cdots+A^n=0.$

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying ...
0
votes
4answers
41 views

On which terms $rank(A)=3$?

$$\left( {\begin{array}{*{20}{c}} a & b & {a + b} \\ {2a} & {a + b} & {a - b} \\ a & a & {2a - b} \\ \end{array}} \right) \simeq \left( {\begin{array}{*{20}{c}} ...
3
votes
2answers
792 views

Matlab code for computing index of matrix

I need help to make Matlab code for computing index of matrix. The index of matrix $A$ is the least non negative integer $k$ such that $\operatorname{rank}(A^{k+1}) = \operatorname{rank}(A^k)$. ...
2
votes
2answers
492 views

Flood algorithm - find polygon containing a given point.

I have some code that represents a set of a set of interconnected line segments in 2D, in pseudo-code it'd be like this: ...
2
votes
1answer
83 views

An inequality on the root of matrix products

Suppose $A$ and $B$ are positive definite (symmetric) real matrices. Is it true that $(AB)^{1/2}+(BA)^{1/2} \geq A^{1/2}B^{1/2}+B^{1/2}A^{1/2}$ ? EDIT: Shown to be false below. Extension sought: ...
0
votes
1answer
53 views

Solve the linear system $Ax=b$ given $x_{1}, x_{2}, x_{3}$ and $b$

Given $x_{1}=(1,1,1)$ and $x_{2}=(0,1,1)$ and $x_{3}=(0,0,1)$, solve $Ax = b$ when $b=(3,5,8)$. Furthermore, what is A? My thought process was as follows: If $Ax_{1}=b$, then $A = ...
2
votes
0answers
53 views

Conjugacy classes of unipotent $\mathbb{Z}\times\mathbb{Z}$ in $GL_3(\mathbb{Q})$

Let $G=\mathrm{GL}_3(\mathbb{Q})$. Now, consider all subgroups in $G$ of the form $\mathbb{Z}\times\mathbb{Z}$ consisting only of unipotent elements (elements whose eigenvalues are all $1$). How ...
6
votes
2answers
164 views

What can be computed by axiomatic summation?

Here are three simple properties one might require of a summation method for divergent series: A stable summation scheme is one in which (assuming also each sums are defined iff the other is) ...
1
vote
0answers
296 views

Minimum L1 norm may not obtain the sparsest solution?

Here is a paper called For Most Large Underdetermined Systems of Equations, the Minimal L1-norm Near-Solution Approximates the Sparsest Near-Solution. However, I did not quite get its definition of ...
3
votes
2answers
153 views

Expressions for Permanent of a Matrix

Given that the permanent of a matrix can be written in a similar form as the determinant, as a sum of permutations of the elements of the matrix, is there also a relationship between the permanent and ...
0
votes
2answers
59 views

Prove the existence of a multiplicitive Inverse

Let $F$ be a field, such that $$F=\{a+b \sqrt{2}\}$$ Such that a and b are rational numbers. Prove there exists a multiplicative identity. I just expanded the product of two elements, and collected ...
3
votes
3answers
2k views

Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular.

I'm trying to prove that statement: Prove that $\lambda = 0$ is an eigenvalue if and only if $A$ is singular. I'm not sure if my proof is totally correct: Suppose that $\lambda = 0$ if det(A) = ...
1
vote
1answer
73 views

Matrix Similarity through Schur's Formula

I am trying to solve the following exercise to prepare for qualifiers. The purpose of this exercise is to get familiar with Schur's formula. (This exercise is apparently very basic). Let $A$ be a ...
5
votes
3answers
75 views

Can non-normal matrices with double eigenvalues never be diagonalized?

Is there a matrix $A$ with $A^TA≠ AA^T$ (non-normality) and double eigenvalue that is still diagonalizable? If $A^TA \neq AA^T$ and $λ_1 =λ_2 = λ$ (double eigenvalue) $\stackrel{?}{⇒}$ not exists ...
2
votes
1answer
576 views

Vector space of real vectors over field complex scalars.

Let $V = \{(a_1,a_2,\ldots,a_n): a_i \in \mathbb{R}\ \text{ for } i = 1,2,\ldots,n\};$ So $V$ is a vector space over $\mathbb{R}$. Is $V$ a vector space over the field of complex numbers with the ...
1
vote
1answer
74 views

Direction of steepest descent and minimization?

I have the following linear function: $min$ 1/2 $<x, x>$ + $r^Tx$ for every x belonging to $R^n$, $r^Tx$ belongs to $R^n$ Now, = $x^TAy$ and A is symmetric positive definite. = $x^TAy$ is ...
0
votes
1answer
107 views

If two nxn matrices have the same (non-zero) determinant, they can be transformed into the same form by gaussian elimination - true or false?

If the determinants are 0, the theorem is obviously false, but what happens otherwise? And if it does not hold, could we add some other conditions to make it true? My motivation for this came to me ...
1
vote
2answers
67 views

Find the Slopes and Y-intercepts.

Im currently doing an Algebra course on a site, and in the test I'm currently on these problems are stablished. What are the slopes and y-coordinates of the y-intercepts of the following lines. ...
0
votes
1answer
24 views

Prove that a sequence of linear maps is bounded iff its matrix representation is bounded.

Let $f$ be an endomorphism of a finite dimensional vector space. We consider the following sequence of maps $(f^p)_p$. $M_B(f)$ is the matrix representation of $f$ in the basis $B$ The following ...
-1
votes
1answer
156 views

Solve nonlinear equations: variables with degree six and degree eight.

Suppose I have two nonlinear equations with two variables $\ell$ and $m$; the variables $\ell$ and $m$ are of degree eight in the first equation and of degree six in the second equation. How it ...
1
vote
2answers
333 views

Gram Determinant equals volume?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly m =< n. In general, for computational purposes, what I have managed to do ...
2
votes
1answer
129 views

eigen values and eigen vector of a matrix [duplicate]

Let $A$ be an n *n matrix all of whose entries are 1. Find all the eigenvalues and eigenvectors of $A$. I have checked for 2*2 ,3*3 matrices and guessing the answer but in general how to show.
1
vote
2answers
346 views

Books on geometric transformations and/or analytic geometry?

I've been looking to expand my knowledge in geometry as it's not covered in my undergraduate curriculum. For some reason I'm repelled by the classical approach (hopefully it will pass) as I feel it's ...
0
votes
2answers
85 views

Is there a linear transformation $\psi: Mat_{3x2}(ℝ) \mapsto ℝ_8[x]$ that is surjective?

Is there a linear transformation $\psi: Mat_{3x2}(ℝ) \mapsto ℝ_8[x]$ that is surjective? $Mat_{3x2}(ℝ)$ is the space of real $3$x$2$ matrices. $ℝ_8[x]$ is the space of all real polynomials up to ...
6
votes
6answers
644 views

Prove $e^x, e^{2x}…, e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$

Prove $e^x, e^{2x}..., e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$ isn't it suffice to say that $e^y$ for any $y \in \mathbb{R}$ is in $\mathbb{R}^+$ Therefore, ...
2
votes
0answers
83 views

Solving a set of matrix equations

There are $k$ matrix equations with the same unknown $\mathbf{X}$: $\mathbf{A}_i(\mathbf{D}_i-\mathbf{X})^{-1}\mathbf{B}_i=\mathbf{C}_i$ where $i=1,2,...,k$. $\mathbf{A}_i$ is a $m\times n$ matrix. ...
0
votes
1answer
39 views

find a span for a polynomial

$$\{ a{x^3} + b{x^2} + ( - a - b)x\} = sp\{ {x^3} - x,{x^2} - x\}$$ How do you find the span of a polynomial? I'd be glad if you could explain what was done in the example above. thanks
2
votes
1answer
189 views

Existence of a non-singular $n-2$ principle minor

Suppose $A$ is a $n\times n$ real symmetric matrix with $|A|\ne0$, and all the principal minors of order $n-1$ have determinant zero. Prove: there is a principal minor of order $n-2$ which is ...
3
votes
0answers
63 views

Find the rank of the matrix

Let $X,Y\in\mathbb R^n$ be two non zero (column) vectors. Let $Y^T$ denote the transpose of Y. Let A = $X Y^T$. What is the rank of $A$ and what is the necessary and sufficient condition for the ...
6
votes
3answers
246 views

If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$

When reading the topic about primary and rational canonical form of matrices I stuck myself on this theorem: The matrices $A,B\in K^{n\times n}$ are similar if and only if their characteristic ...
3
votes
1answer
109 views

Linear algebra question on rank and null space

This is an exercise from linear algebra and optimization by Gill, I do exercises to be prepared for my final exam and these are not homework questions! Exercise $\mathbf{6.1.}\,$ Consider the ...
3
votes
0answers
206 views

Jordan canonical forms determined by a minimal polynomial

Find the Jordan canonical forms of all $9\times 9$ matrices over $\mathbb{C}$ with minimal polynomial $x^2(x-3)^3$. My method: each factor of the minimal polynomial corresponds to a type of Jordan ...
0
votes
1answer
76 views

System of 2 linear q-difference equations with singular matrix

I would like to solve the following algebraic linear system of q-difference functional equations: \begin{cases} ...
1
vote
1answer
39 views

linear algebra : matrix decomposition

Let $X \in \mathrm{Mat}_{n \times p}(\mathbb{R})$ a matrix such that $\mathrm{rank}(X)=p$. Let $S = \mathrm{I}_{n} - X \big( X^{\top} X \big)^{-1} X^{\top}$ be the orthogonal projection on $\big( ...
1
vote
1answer
133 views

What is and how can I find an orthogonal component?

I have the following task : Find the orthogonal projection and orthogonal component of vector $ \overline x $ relatively to linear subspace generated by vectors $ \overline a_{1}, ...
2
votes
3answers
189 views

Characterize the polynomial bijections from $(0,1)$ to$ (0,1)$

That is, show which polynomials, taken as functions, truncated to $(0,1)$, biject with $(0,1)$. I know for the increasing case, it is equivalent to (the derivative is greater than or equal to $0$ on ...
0
votes
1answer
174 views

How many moves (shifts) are needed to sort an unsorted sequence of numbers $1$ to $n$ in ascending order?

I have the LUP decomposition of a matrix. The determinant can be found from the formula: $$\det(A) = \det(P^{-1}) \det(L) \det(U) = (-1)^s \left( \prod_{i=1}^n l_{ii} \right) \left( \prod_{i=1}^n ...
3
votes
2answers
9k views

Formula to project a vector onto a plane

I have a reference plane formed by 3 points in R3 – A, B & C. I have a 4th point, D. I would like to project the vector →BD onto the reference plane as well as project vector →BD onto the plane ...
1
vote
1answer
81 views

confusion about rank and nullity of matrix and rank-nullity theorem

I can see that rank + nullity = number of columns of the matrix. Does that mean the dimension of matrix is the number of columns? Isn't dimension of a $m\times n$ matrix $mn$? Thanks.
1
vote
0answers
28 views

Vectors evolving by $\frac{\partial \vec{x}}{\partial t}=\vec{U}(\vec{x})$: maximize effect of perturbation.

Suppose we have a finite dimensional real vector space $V$ equipped with a norm $\|\cdot\|$ which is given by $$\|\vec{x}\|^2 = \vec{x}^TX\vec{x},$$ where $X$ is a matrix and $x\in V$ is in ...
0
votes
2answers
69 views

Rank of a 2 x 2 matrix

Prove that the rotation matrix is invertible. Let $$ \begin{pmatrix} \cos 2\pi t & \cos \frac{\pi}{6}t\\ \sin 2\pi t & \sin \frac{\pi}{6}t\\ \end{pmatrix} $$ What is the rank of the ...
3
votes
1answer
78 views

Relationship between a finite codimensional subspace of dual space and the annihilator

Notation: $X$ is a banach space, $X'$ is the dual space to $X$. When $V \subset X'$, we write $\ker V = \cap_{l \in V} \ker l$, and when $W \subset X$, we write $ann \; W = \{l \in X' \mid l(w) = 0 ...
4
votes
2answers
101 views

Recursive determinant of given matrix in $\mathbb{R}^{n\times n}$

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
2
votes
2answers
233 views

Solving equations involving both matrix and three-index unknowns

Suppose I have two equations where the two unknowns and constants are square matrices. That's easy to solve since I can invert the matrices. But what if I have something like ...
0
votes
3answers
682 views

Quadratic Function whose graph contains the points:

A quadratic function is a function of the form $y=ax^2+bx+c$ where a, b, and c are constants. Given any 3 points in the plane, there is exactly one quadratic function whose graph contains these ...
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1answer
36 views

$S:\mathbb{R}^n\to\mathbb{R}^n$ be given by $v\mapsto \alpha v$ for a fixed $\alpha\ne 0$ (real)

$S:\mathbb{R}^n\to\mathbb{R}^n$ be given by $v\mapsto \alpha v$ for a fixed $\alpha\ne 0$ (real),$T$ be another linear map such that $\beta=\{v_1,\dots,v_n\}$ is a set of linearly indipendent eigen ...
1
vote
0answers
607 views

What do you call the product of a matrix's diagonal elements?

The trace of $A$, an $N\times N$ matrix, is $\sum_{i=1}^N A_{ii}$. What do you call $\prod_{i=1}^N A_{ii}$?
2
votes
1answer
63 views

Finding linear transformations with basis values

I need some help "proof-reading" the following exercise and making sure there are no mistakes. Plus, I need help on b)! Consider the vector space $V=\mathbb{R}_3[x]$. Let $f \colon V \to V$ be the ...
4
votes
1answer
616 views

All Invariant Subspaces of a Linear Transformation

I got this problem: Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3 and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ ...
1
vote
1answer
46 views

Solving a system of nonlinear equations

Consider a machine that operates using the following equation: O $_3\ _×\ _1 = $ X $_3\ _×\ _3 ×$ [I $_3\ _×\ _1$ − Y $_3\ _×\ _1 $] + Y $_3\ _×\ _1,$ where I $_3\ _×\ _1$ is the input and O $_3\ ...