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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8
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Matrices that are not diagonal or triangular, whose eigenvalues are the diagonal elements

I want to learn about matrices whose diagonal elements are the eigenvalues... but the matrix is neither diagonal nor triangular. Is there a term for such matrices, and have they been researched?
3
votes
1answer
38 views

The maximal rotation matrix

Let's consider two numbers calculated for a rotation matrix which are: $s_e=$ the sum of all entries of a matrix $s_a=$ the sum of absolute values of all entries for a given matrix. It ...
0
votes
2answers
28 views

Problem in finding examples of linear operators.

Find the example of two linear operators $T$ and $U$ such that $TU = O$ but $UT \neq O$. But I fail to find out proper example.Please help me in finding the example.Thank you in advance.
1
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1answer
573 views

Transformation matrix from a translated-rotated coordinate system to the general coordinate system

In Figure 1, suppose $XYZ$ (in black) as my general coordinate system and $X'Y'Z'$ (orange) as another system with parallel axes respect to $XYZ$. Consider $xyz$ (green) is my 3rd coordinate system ...
0
votes
2answers
19 views

how to know if a set of arbitrary vectors are a basis?

So, if we're given that $\{x,y,z,w\}$ is a basis of $\Bbb R^4$, how do we show that $\{x+w, y+w, z+w, w\}$ is also a basis of $\Bbb R^4$? I know that for a set to be a basis, it has to be linearly ...
1
vote
2answers
30 views

If A is positive definite (but not necessarily symmetric) can you decompose it?

If A is a $2 \times 2$ matrix that is positive definite but may or may not be symmetric, does there exist another matrix B such that $A=B^TB$?
0
votes
2answers
16 views

Given the matrix, find a matrix such that

Given $T(\begin{bmatrix}1\\-2\end{bmatrix}) = \begin{bmatrix}3\\10\end{bmatrix}$ $T(\begin{bmatrix}-2\\-1\end{bmatrix}) = \begin{bmatrix}-1\\-5\end{bmatrix}$ Find a matrix such that: $T(...
0
votes
1answer
14 views

Using matrix operations to wrap arrays

I'm coding with Maple and I need an efficient way to wrap arrays of numbers to different dimensions. For example, let $A$ be the 3x5 matrix listing the numbers 1-15 in order, and $B$ be the 5x3 ...
3
votes
2answers
30 views

How may I use a 3x3 matrix to simulate a larger square matrix?

I am using a game engine where the library only provides 3x3 matrices with the multiplication and inverse operation. I could build my own matrix library to provide larger matrices, but it would be ...
3
votes
1answer
37 views

Show that $A\varphi_j=\left<A\varphi_j,\varphi_j\right>\varphi_j$ and $A^*A\varphi_j=s_j(A)^2\varphi_j$ for all $j$

Let $A$ be a bounded linear (compact) operator acting on a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\...
4
votes
2answers
76 views

Prove that if $y=(y_1, \ldots, y_n)$ is such that $y_1a_1 + \cdots + y_na_n = 0$, then $∀x ∈ \mathbb{R}^n$, $Ax · y = 0$

I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let $A$ be a $n\times n$ matrix and let $a_1,\ldots,a_n$ be the rows of $A.$ Suppose $y=(y_1, \ldots, ...
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votes
1answer
67 views

haw can prove the a map is linear

Let $E$ and $F$ are two vector space of finite dimension on the same field $k$, so we can assume $E = k^n $ and $F=k^m$. Let $f$ be an application from $E$ to $F$ explicitly donated. The question is ...
0
votes
1answer
30 views

Extension of mapping of subset to homomorphism

Is the following proposition true? Let $V$, $W$ be vector spaces over some field $F$ and let $S_v \subset V$. Then if every mapping $f:S_v \to W$ can be uniquely extended to homomorphism $g:V\to W$, $...
0
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0answers
14 views

A non-strict inequality on skew symmetric matrices

As we know that skew-symmetricity means $A=-A^\top$ where $A\in\mathbb{R}^{n\times n }$. But recently I came across an inequality that states, $A+A^\top\preceq0$ can also be considered as an ...
1
vote
1answer
27 views

Vector space endomorphisms of sequence space

Suppose we are working in the sequence space $K^\mathbb{N}$, defined as follows: Let $K$ be the field either of real or complex numbers. We denote $K^\mathbb{N}$ the set of scalars $(x_n)_{n\in\...
3
votes
1answer
38 views

Is there an $\alpha\in\mathbb{R}^m$, such that $\alpha_i > 0$ and $A\alpha\in S$?

$A$ is a real $n\times m$ matrix and set $S\subseteq \mathbb{R}^n$ is defined as $$S = \{(x_1,\dots, x_n)\in \mathbb{R}^n\mid \forall(i,j)\in I.\; x_i< x_j\}\text{,}$$ where $I$ is a possibly empty ...
-1
votes
0answers
23 views

MAthematical notation for sorting submatrix and replacing it back

I need help in expressing the following paragraph in mathematical form as much as possible. I have a matrix $A$ which is $N\times M$. For each element of $A$, $A(i,j)$, I consider a submatrix of $A$ ...
1
vote
1answer
28 views

Calculating block diagonalization / canonical bases with linear optimization?

Edit Even though I have started answering my own question I am still eager to hear any feedback and new ideas. So feel free to tell me if you come to think of anything. In Linear Algebra there are ...
1
vote
4answers
90 views

How to prove $I-BA$ is invertible [duplicate]

Show that $I-BA$ is invertible if $I-AB$ is invertible. And also, we have to prove that eigenvalues are same for $AB$ and $BA$ Till now, I used the equation $(I-AB)(I-AB)^{-1}=I$ which gives $(I-AB)...
2
votes
1answer
42 views

Column sums of $A$ from column sums of $A A^T$?

Let $A$ be an invertible matrix. Is it possible to infer anything about the column sums of $A$ by precisely knowing the column sums of $A A^T$? What if we impose some restriction on the $A $ that we ...
2
votes
2answers
30 views

Find vectors u and v such that W = Span{u,v}

Let $W$ be the set of all vectors of the form $\begin{bmatrix}s-t\\2s+t\\0\\t\end{bmatrix}$ Find vectors $u$ and $v$ such that $W =$ Span{$u,v$} How can I do this? Any advice woulds be ...
0
votes
1answer
30 views

decomposition of a square matrix

my professor uses this decomposition all the time and I don't know why it's allowed. he told me it's true for any square matrix (I assume any real matrix). why can I decompose any square matrix A, to:...
0
votes
3answers
35 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
0
votes
1answer
20 views

I need a simple equation to measure a efficiency of attempts correction

I have a process where the user need correct an invalid information in your registry within a maximum number of attempts. The closer he gets this maximum number, the worse your rate. For example: ...
2
votes
3answers
46 views

Explain why the columns of a 3x4 matrix are linearly dependent

Explain why the columns of a $3 \times 4$ matrix are linearly dependent I also am curious what people are talking about when they say "rank"? We haven't touched anything with the word rank in our ...
0
votes
1answer
22 views

Matrix Rotations and Enlargements Help

Matrix $M$ is given as \begin{bmatrix}3&-{\sqrt 7}\\{\sqrt 7}&3\end{bmatrix} I then am asked to describe the transformation, you are also told dis an enlargement followed by a rotation and you ...
0
votes
0answers
24 views

Null spaces and their dimensions. how to decide the dimension of the null space

Here is a quote from a textbook. Within four dimensional space ofa ll possible vectors x the solutions to $$Ax=0$$ form a two dimensional subspace - the nullspace of A In this specific A we ...
1
vote
1answer
2k views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
3
votes
0answers
37 views

Creating a tight frame of $\mathbb{R}^{n}$ when already knowing some of its vectors.

I'm wondering whether or not it's possible to start with a matrix $S\in\mathbb{R}^{m\times mn}$, $m<n$, and add rows to it so that the columns of the resulting matrix form an orthogonal system of ...
0
votes
1answer
21 views

is I both a lower triang enad upper triangle ( Also proving L1=L2 )

First part of the question is $$ A= L_1D_1U_1\\ A = L_2D_2U_2\\ Prove\\ L_1= L_2\\ D_1 = D_2 \\ U_1 = U_2 \\ $$ My attempt seems correct but not quire sure whether it's mathematically constructed. $$...
1
vote
3answers
56 views

is there a non unit real matrix satisfied $A^n=I$?

If A is a real matrix and $A^{2016}$ is a symmetric positive definite matrix , prove that $A$ also is a symmetric positive definite matrix I wonder if this property is wrong and so I came up with ...
0
votes
2answers
16 views

Direct vs Iterative solvers choice

Is there any other reason except “the big size of matrix” that makes me prefer the use of iterative solvers than direct ones, for (linear algebraic systems)? Thanks
3
votes
1answer
33 views

All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
2
votes
2answers
29 views

Are a uniformly random polynomial's roots are distributed uniformly in the field?

Assume we have a $\mathbb{F}_p$, where $p$ is a large prime (e.g. 128-bit value). We define all polynomials over the field, and pick a polynomial,$P(x)$, of degree $d$, where the polynomials' ...
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0answers
25 views

Help in this proof from Hoffman and Kunze's Linear Algebra book

I'm reading Hoffman and Kunze's Linear Algebra book and on page 177 they stated and proved the following theorem: It's a big proof which I didn't understand only a very little part of it: I ...
0
votes
1answer
10 views

Eliminate asymptote using projective transform

I have a well-behaved curve $f:\mathbb{R}\rightarrow \mathbb{R}^2$ which has exactly one linear asymptote passing through points $p$ and $q$ in $\mathbb{R}^2$. I would like to find a projective ...
0
votes
2answers
33 views

How come associative law of matrix multiplication won't work when permutation matrices come in. Which is the case for some

if $$x=y$$ explain why $$Px=Py$$ I believe this part is very since when we do $$P^{-1}Px = P^{-1}Py$$ from here $$x=y$$ But the other part of the question seems much more confusing then $$(Px)^...
0
votes
3answers
28 views

Inverse of matrix with very structured submatrix

Does this matrix admit an easy analytic expression for its inverse? $$\begin{bmatrix} a_1 & 0 & 0 & 0 & 0 &0&\dots&0 \\ a_2 & 1 & -b & 0 & 0&0&\...
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votes
2answers
37 views

Solve three equations for three unknowns. [duplicate]

So I have the following three equations which I do not know how to solve: -D * x - E * y = A + (R * D) E * F * x - D * F * y - G * z = B - (R * E * F) E * G * x - D * G * y + F * z = C - (R * E * G)...
0
votes
3answers
53 views

Showing A is not invertible

$$ A= \begin{bmatrix} 2 & 1 & 4 & 6 \\ 0 & 3 & 8 & 5 \\ 0 & 0 & 0 & 7 \\ 0 & 0 & 0 & 9 \\ \end{bmatrix} $$ We are asked to show A is not invertible ...
0
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0answers
10 views

What is the optimized Time complexity of Cholesky decomposition

Is there any algorithm for Cholesky decomposition that has complexity O(n^a) where a < 3? I know there are some algorithms to be better than n^3 for matrix multiplication, not sure about Cholesky, ...
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0answers
17 views

Eigenvalue of multiplicity k of a real symmetric matrix has exactly k linearly independent eigenvector

If A is an nxn real symmetric matrix then A is diagonalisable. In other words, If A is a symmetric nxn matrix, then there exists an orthogonal matrix $P$ such that $P_{-1}AP=P_{T}AP=D$, a diagonal ...
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0answers
11 views

alignment of two sets of vectors

I have a maximisation problem to do with aligning two ordered sets of 3D unit vectors. I want to apply the same rotation to all the vectors in one set so that they are in closest alignment with those ...
2
votes
1answer
56 views

Problem about linear algebra [duplicate]

Suppose we have two $n \times n$ square matrices A and B such that $AB=BA$. It is known that A, B and AB all have n distinct eigenvectors that is a basis of $\mathbb{C}^n$. Can we then show that there ...
0
votes
3answers
27 views

How to find other basis of polynomials of degree three or less? [on hold]

How can i find a basis of polynomials of degree three or less, which is other than $\{1,t,t^2,t^3\}$ ?
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votes
2answers
20 views

Finding inverse by elimination

Find the inverse of the matrix $A$ below by elimination on [A I] By expanding the matrix into an alternating matrix. $$ A= \begin{bmatrix} 1 & -1 & 1 & -1 \\ 0 & 1 & -1 & 1 \\ ...
2
votes
1answer
542 views

MATLAB determining elementary matrices for LU decomposition

I am confused by this question I am studying for MATLAB practice.
2
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1answer
54 views

Solving the linear system $XL + L^TX = M$ efficiently

I'm wondering of an efficient way to solve the following system for the symmetric matrix $X$, given a positive semi-definite matrix $S$ and any matrix $M$: $$ LL^T = S $$ $$ XL + L^TX = M $$ $$ (XL) + ...
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2answers
20 views

Choosing independent entries in a symmetric matrix

So, the question is how many entries can be chosen indepently in a symmetric matrix of order n? 2) How many entries can be chosen indepently in a skew-symmetric matrix $$ K^T=-K $$ of order n. The ...