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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Rank of the matrix

Let ${\bf A} $ is a matrix that construct with coordinate of the $n$ distributed points in a two dimensional domain like follow: $${\bf A }=\begin{pmatrix} 1&x_1&y_1\\ 1&x_2&y_2\\ \...
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32 views

In the context of linear algebra, is it possible for a vector space or a subspace to have a finite number of elements? [duplicate]

A vector space must satisfy closure under addition and multiplication. Sorry if this is obvious but does that mean that, assuming the normal rules of arithmetic and excluding the trivial examples like ...
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1answer
33 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 ...
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Solve $\max \mathrm{sum}(AXB \geq \gamma), X \in \{0,1\}^{N \times N}$

I have a problem to find the best permutation matrix $X \in \{0,1\}^{N \times N}$, so as to maximize the number of elements in $AXB$ which are above a certain positive number $\gamma$. In other ...
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1answer
947 views

Why does SVD provide the least squares solution to $Ax=b$?

I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=b$, ...
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1answer
16 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 $W$ ...
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1answer
47 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 ...
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4answers
32 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 ...
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1answer
50 views

What exactly are operations involving tensors… In terms of their indices

So I have heard that tensor operations involve the faces of the rectangular prism. These are matrices right, and different properties of those matrices say things about the tensor? Could someone ...
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19 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 ...
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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 ...
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+50

Is it Possible to Show that the Determinant of a Symplectic Matrix is 1 Using Induction?

We have for a $2 \times 2$ matrix $A$ that $A$ is symplectic if and only if $\det A =1$. Is there any way to use this fact as the base for an inductive proof of the fact that the determinant of any ...
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1answer
73 views

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

I'm wondering whether or not there's an optimal way for adding rows to a given matrix $S\in\mathbb{R}^{m\times mn}$, $m\leq n$, so that the columns of the resulting matrix form an orthogonal system of ...
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1answer
16 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 ...
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2answers
659 views

Linear Algebra: Change of Basis

Let $A[a,b,c]$ and $B[d,e,f]$ be two non-standard bases. I have to find the $3\times3$ matrix that will convert a vector defined in terms of $A$ to $B$. My solution is: Let's assume a standard basis ...
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24 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=\...
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1answer
14 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 ...
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1answer
31 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 ...
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1answer
29 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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4answers
99 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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0answers
18 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 & \...
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3answers
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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} ...
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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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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 ...
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361 views

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?
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8 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 ...
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19 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 ...
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2answers
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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)...
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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^...
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4answers
31 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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1answer
52 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 ...
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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.
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1answer
574 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 ...
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2answers
22 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 ...
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2answers
37 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 $A=B^TB$?
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2answers
26 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(...
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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 ...
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2answers
31 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
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1answer
39 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,\...
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2answers
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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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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 ...
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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 ...
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1answer
36 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\...
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1answer
40 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 ...
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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 $A$ ...
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1answer
44 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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2answers
33 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 ...
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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:...
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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}...