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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63 views

Converse of Schur's Lemma in finite dimensional vector spaces

I am trying to prove (or disprove) the converse of Schur's Lemma in finite dimensional vector spaces. I am not sure if it holds in this case, but I have tried to apply the idea that proves it in ...
1
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0answers
36 views

Moore-Penrose pseudoinverse and Linear relations

I recently came across this website called Graphical Linear Algebra. I feel like there's a lot of insight there, but it's too monolithic for me to be able to extract it by skimming. Episode 27 is ...
0
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4answers
44 views

If $Y = X\beta$ are a system of linear equations and that $X$ is NOT full rank. Is this system under or over determined?

Suppose I have a system of linear equations, $Y = X\beta$, where $Y$ is a $n$ by $1$ matrix, $X$ an $n$ by $n$ matrix, and $\beta$ a $n$ by $1$ matrix. Suppose that I know what $Y$ and $X$ are, and ...
2
votes
1answer
18 views

Can we get $\|A^\dagger x-B^{-1}x\|_2\leq \epsilon \|B^{-1}x\|_2$?

In the question: the $A\in R^{d\times d}$ is positive semi-definite, $B\in R^{d\times d}$ is positive definite, $x\in R^d$ is a vector, and $\epsilon$ is a variable that may depend on $A$, $B$ and ...
0
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1answer
17 views

Can a scalar multiple for a linearly dependant vector be undefined?

Let $A = \begin{bmatrix}a & b & c \\ 2a & 2b & 2c \\ e & f & g\end{bmatrix}$ where $R_{3}$ is linearly independent of $R_{1}$, Clearly, $(-2 , 1, 0)$ will be set of scalars ...
0
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0answers
22 views

Show that this vector is not a function of $\tau$

I have a variance matrix given by: $\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{'}+\Omega$ where $\Omega=\left(\begin{array}{cccc} \sigma_{\varepsilon}^{2}\psi\left(\tau_{1}\right) & 0 & \...
1
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1answer
56 views

Spectral theorem for diagonlizable matrices

For a diagonalizable matrix $\textbf A_{n \times n}$ with spectrum $σ(\textbf A)=\{\lambda_1, \lambda_2,..., \lambda_k\}$ we have matrices $\{ \textbf G_1, \textbf G_2,..., \textbf G_k \}$ such that: ...
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0answers
26 views

Why does power iteration generate almost dependent vectors?

On the Wiki page for Krylov subspaces: https://en.wikipedia.org/wiki/Krylov_subspace it states given a matrix $A$ and vector $b$, that the vectors $b, Ab, A^2b, A^3b, ...$ "soon become almost linearly ...
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3answers
31 views

Solve for $X$ in matrix equation

How can I solve for $X$ in this matrix equation? $$\begin{bmatrix}-3&-8\\-9&5\end{bmatrix} X + \begin{bmatrix}4&-7\\3&-2\end{bmatrix} = \begin{bmatrix}5&8\\-1&-1\end{bmatrix} ...
2
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0answers
32 views

Inner-product on skew-hermitian matrices

Let $$\mathfrak{u}(n)=\{X\in M(n,\Bbb C):X+X^*=0\}$$ where $X^*$ is the conjugate transpose. Then, $\mathfrak{u}(n)$ is a real vector space. Problem. Show that $\langle X,Y\rangle=\...
-1
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2answers
37 views

Example of a degenerate bilinear map?

I seek an example of a nonzero $\Bbb{R}$-bilinear map $f:V\times V\rightarrow W$ on a vector space $V$ (s.t: $\dim V<\infty$, $\dim W<\infty$) such that it is degenerate map, where $V$ and $W$ ...
0
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0answers
24 views

How to estimate the product of the $k$ largest eigenvalues of a matrix

Now I have a question which let me to prove that the product of the largest $k$ singular values of a real matrix is always larger than the one of $k$ largest eigenvalues. For $k=1$, I use the ...
0
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0answers
11 views

Spectral norm of the matrix derivation

I understand one possible way how to derive induced norm of symmetrix matrix M, i.e. $sup |M \tilde{x} |$, s.t. $|\tilde{x}|=\tilde{x}^T\tilde{x}=1$ (i.e. $\tilde{x}$ is lie in unit sphere) Here is ...
0
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0answers
24 views

Dimension from inequalities

Let $S$ denote $(x,y,z)\in \mathbb R^3$, which satisfies the inequalities: $$x - 2y + z \leq 1$$ $$2x + 2y - z \leq 5$$ $$-2x + y + z \leq 4$$ $$x \geq 1$$ $$y \geq 2$$ $$z \geq 3$$ How do I ...
1
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0answers
14 views

Using Fixed point iterations for solving system of linear equations

Given a system of $n$ linear equations $$ x_i=\sum_{k=1}^{n}a_{ik}x_k+b_i \quad i=1,2,...,n$$ I'd like to employ the fixed point iteration method to find $x_i$. The fixed point iteration define $$ x_i^...
0
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4answers
34 views

whether set of 3D vectors span space {(x, y, z) | x + y + z = 0}

Consider the set of (column) vectors defined by $X = \{x \in R^{3} | x_{1} + x_{2} + x_{3} = 0\}$, where $X^{T} = [x_{1}, x_{2}, x_{3}]^{T}$ , I need to prove whether(or not) given vectors, $[1, -1, 0]...
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0answers
15 views

Solving $I^* = \arg\min_{I'} \left( \|\phi_\ell(I) - \phi_\ell(I')\|_2^2 + R(I') \right)$ with gradient descent

I am trying to create the results from this a paper that is trying to understand the types of features a convolutional neural network is learning to recognize. I don't think understanding ...
0
votes
2answers
30 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
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2answers
24 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 ...
3
votes
2answers
34 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 ...
1
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2answers
42 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 $A=B^TB$?
2
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1answer
45 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 ...
0
votes
1answer
59 views

Extension of mapping of subset to homomorphism

Is the following proposition true? Let $V$, $W\neq0$ 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 ...
2
votes
1answer
45 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 ...
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1answer
33 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 $A$ ...
2
votes
1answer
40 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\...
2
votes
2answers
35 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
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0answers
30 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 ...
0
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3answers
40 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}...
2
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3answers
49 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
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1answer
22 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: ...
0
votes
2answers
30 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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3answers
59 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
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2answers
18 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
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0answers
28 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 ...
3
votes
1answer
57 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 ...
1
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3answers
33 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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0answers
13 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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votes
2answers
48 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
1answer
36 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
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2answers
36 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
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1answer
29 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 ...
2
votes
2answers
33 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
18 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
4answers
111 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)...
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3answers
28 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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2answers
27 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 \\ ...
0
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1answer
27 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. $$...
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2answers
21 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 ...
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votes
3answers
55 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 ...