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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If the product of two matrices, $A$ and $B$ is zero matrix, prove that matrices $A$ and $B$ don't have to be zero matrices

I can give an example where product of two non-zero matrices is zero matrix, $$ A= \begin{bmatrix} 3 & 6 \\ 2 & 4 \\ \end{bmatrix} $$ $$ B= ...
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0answers
27 views

Let $V$ be a $K$-vector space, let $f:V\to V$ be a linear operator, show equivalence between $f$ is injective, surjective and bijective

I have some issues understandig a proof of the following theorem. Let $V$ be a finite-dimensional $K$-vector space and $f: V \to V$ a linear operator. Then the following are equivalent: ...
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2answers
40 views

Bases s.t. matrix for $T: V \rightarrow W$ is diagonal

Let $T: V \rightarrow W$ be a linear map between finite-dimensional vector spaces. Show that there exist bases ${e_i}$ of $V$ and ${f_i}$ of $W$ such that the matrix of $T$ has entries ...
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2answers
32 views

eigenvalue problem for $n$ dimensional matrix

Lets say I have $n$-dimensional matrix $$ \hat T = \begin{pmatrix} e^{Y} & 1 & \cdots & 1 \\ 1 & e^{Y} & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots\\ 1 & ...
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0answers
35 views

Symmetric Matrices Sign of pivots

It is said that symmetric matrices have equal number of positive eigen values as that of positive pivots. But when finding the reduced row echelon form, we are free to multiply a row by a scalar ...
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2answers
61 views

Vectors as linear combinations

I want to express v=$(\sqrt2,1/2)$ as linear combination of p=(1,2) and q=(2,1). My answer: $(\sqrt2,1/2)$=c(1,2)+d(2,1). So solve for c and d: $\begin{pmatrix} \sqrt2 \\ 1/2 \end ...
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1answer
52 views

How to derivative the linear equation of matrix

I have the equation as $$F(w,x)=\sum_{i=1}^{N}\int_{x \in \Omega} \left ( Y(x)-w^TA(x)\right)^2u_i(x)dx$$ In which, $w$ is column vector that independent on $x$, denotes $w=[w_1,w_2...,w_M]^T$ $A$ ...
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1answer
60 views

Prove $ \DeclareMathOperator{\trace}{trace} \DeclareMathOperator{\adj}{adj} {p_{A}(t) = \trace(\adj(tI − A))} $

Given $A$ is a $n \times n $ matrix. If ${p_{A}(t)}$ denotes the characteristic polynomial of $A$, I need to prove that $$ {p_{A}(t) = \trace(\adj(tI − A))} $$ I know that ${{\det(A)\,I = ...
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3answers
567 views

differentiable functions and vector spaces

I am having trouble understanding where to start with the following question: Let $F$ be the set of all differentiable functions on $[a,b]$. Show $F$ is a vector space with the standard operations. ...
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1answer
32 views

Singular values and singular vector pair for the matrix

What are the singular values and singular vector pairs for a matrix $1_{1\times q}\otimes w_{p\times 1}$? Here $1_{1\times q}$ is the row vector of all ones, $w_{p\times 1}$ is an arbitrary column ...
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1answer
73 views

$d \times d \times d$ tensor rank vs $d \times d$ tensor rank

I am trying to understand rank of a $d \times d \times d$ tensor. The way that I understand the $d \times d$ case is that a rank $r$, $d \times d$ tensor is a tensor that can be written as the sum of ...
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1answer
21 views

Given a matrix $H$, how can I find the “eigenmatrices” $E_\alpha$ such that $[H,E_\alpha]= \alpha E_\alpha$

This is just an Eigenvector problem, but I'm not sure how to tackle it. If we have instead of ordinary matrix multiplication the commutator $[A,B]=AB-BA$ and instead of an eigenvector $\vec v$ we're ...
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1answer
112 views

Verifying orthogonality between two binary sequences

I have studied that for orthogonality to exist between two binary sequences: [Number of bit agreements - Number of bit disagreements]/sequence length=0 Eg, for an orthogonal matrix X given by: ...
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2answers
39 views

Jänich linear algebra, Question 2.3 solution clarification

Let $ U_{1} $ and $ U_{2}$ be vector subspaces of a vector space $ V $. Prove if $ U_{1} \cup U_{2} = V $, then $ U_{1} = V $ or $ U_{2} = V $ or both. Attempt: $ U_{1} \cup U_{2} = V $ ...
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1answer
128 views

L2 norm of an inverse of a sum of matrices

I am trying to take the L2 norm of the following expression: $-(H^{-1} + bI)^{-1}v$, where $H$ is a psd matrix, b is a scalar, and $v$ is a vector. In particular I am having trouble with the first ...
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2answers
743 views

Prove that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^T$.

Prove that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^T$. I'm stucked here, i've approached the problem by looking at $\det(A-\lambda ...
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3answers
59 views

Finding when list of numbers reach periodicity given known values

I'm trying to figure out when numbers reach "periodicity" given known values. I've included an example below with image: I have known sizes (100, 75, and 50) that I would like to know how many times ...
2
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1answer
66 views

show that a matrix is invertible

Let $A$ be an $n \times n$ matrix such that $|a_{ii}|>\sum_{j=1,j\neq i}^n|a_{ij}|$ for each $i$. Show that $A$ is invertible. $(complex matrix) The straight forward way is to show that the ...
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0answers
34 views

Find the term p with matrix .

how I can solve this. Find $p$. Let $x,y,z,p\in\mathbb{R}^{n}$ and $a,b,c$ are constants, such that $$2\langle p,x-y\rangle=(\vert\vert x\vert\vert^2-\vert\vert y\vert\vert^2)-(a^2-b^2)$$ $$2\langle ...
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1answer
16 views

Proving that $L_{22}L_{22}^T=S$ is the Schur complement of a cholesky factorization

Let $A$ be an $n+m \times n+m$ symmetric positive definite matrix. $A=\begin{bmatrix}A_{11} & A_{12}\\ A_{12}^T & A_{22}\end{bmatrix}$ where $A_{11}$ is an $n \times n$ matrix, $A_{12}$ is an ...
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0answers
38 views

Decomposing a matrix as the product of rotations

I'm reading an article about joint diagonalization algorithms. The article states without proof that any nonsingular $n \times n$ matrix $Q$ can be decomposed as \begin{align*} Q = \prod_{1 \leq p ...
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1answer
207 views

Proving wedge product is associative

Fix a real vector space $V$ of finite dimension. Let's denote by $\Lambda^p(V)$ the vector space of $p$-forms on $V$ (i.e. alternating $p$-tensors). Then we have the product $\wedge : \Lambda^p(V) ...
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0answers
115 views

3D projection onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
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2answers
73 views

QR-decomposition

I am doing an exericse from my Linear Algebra text-book and my task is to find a QR-decomposition Find the QR-decomposition of the matrix $A=\begin{bmatrix} 1 & -1 & 4 \\ 1 & 4 ...
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1answer
85 views

What exactly is isomorphism?

I'm studying linear algebra, and i know well what isomorphism is. But i have a problem to understand this concept. Here is an example for explaining my problem explicitly. If $S$ is any ...
2
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0answers
100 views

Conjecture on a graded ring

Consider $B^{(n)}=\mathbb{F}_2[X_1,\dots,X_n]/(X_1^2,\dots,X_n^2)=\bigoplus_{i=0}^nB_i^{(n)}$, where $B_i^{(n)}$ is the space of homogeneous elements of degree $i$. Notice that ...
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1answer
31 views

orthogonalization of a matrix to another matrix

I have an equation $V^T*M*V=E$ so I need to perform an operation on $V$ say $F(V)=V'$ which leads to $V'^T*M*V'=I$ in which $I$ is identity matrix. what is that operation in linear algebra?
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1answer
37 views

Dumbing down 2d cooridinates between two known points

I've searched using various terms and Calculate Points Between Two Points, Incremental Cartensian Coordinates Betwwen Two Known Coordinates and Calculating the x, y coordinate a set distance between ...
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1answer
62 views

Conditions to preserve Laplacian matrix

Let $L$ be a Laplacian matrix $L=D-A$ where $D$ and $A$ are the degree and the adjacency matrices. It is known that $L$ has (among others) the properties: $L=L^T$, $L\geq 0$ and $L1_n=0$, where $1_n$ ...
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1answer
96 views

Matrix integration by parts

It seems to me that the integration by parts rule carries over simply to the matrix case. This can be seen by applying: $(AB)' = A'B + AB'$ and then integrating for square (time dependent) complex ...
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1answer
71 views

For linear transformation $f: V\longrightarrow W$, $\dim R(f) + \dim \ker(f) = \dim V$.

Just starting linear algebra. For every linear transformation $$f: V \longrightarrow W.$$ $\dim R(f) + \dim \ker(f) = \dim V$ Is this correct? $f(x)=2x$ The range of $f$, $R(f)= R_1$ dimension of ...
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2answers
58 views

Proving the following formulae

I need to the following formula: let $g,h\in SL_2\mathbb{R}$, clearly $[g,h]\in SL_2\mathbb{R}$ since its determinant is one. Reading a publication I found the following equality: ...
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0answers
12 views

3D Linear-geometry with coordinates

Truncated pyramid has a smaller opening with sides ABCD, and a bigger opening with sides FGHE ( where F is o top of A, G on top of B, H on top of C and E on top of D). This figure has 3D coordinate ...
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0answers
31 views

SVD for square matrix [duplicate]

I already know the concept of SVD applyed on an mxn matrix. Eigen vectors can't exist for a non-square matrix, but singular-vectors can. My question is: does SVD on a square matrix relate to ...
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2answers
90 views

Determinant of M [closed]

How to find the determinant of the $n\times n$ matrix $M$, whose all the entries are zero except 1st row, 1st column and diagonal entries: $$M= \begin{bmatrix} -x & a_2 & a_3 & \cdots ...
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0answers
30 views

Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function ...
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1answer
141 views

Reducing a pair of indefinite quadratic forms to the canonical form

Assume $A, B$ being a pair of symmetric matrices over reals. Let $$ \varphi_1(x) = (x, Ax)\\ \varphi_2(x) = (x, Bx). $$ There's a well-known result that if $A > 0$ then the pair of forms can be ...
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1answer
82 views

Finding a generating set of vectors

I want to solve the following task: Find the minimal generating set (german: "minimales Erzeugendensystem") for the set S: S = { $\begin{pmatrix} 1 \\ 1 \\ 0 \\1\\1 \end{pmatrix}$, ...
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0answers
23 views

Can the transposition of an arbitrarily-sized matrix be broken up to smaller transpositions?

I'm working with binary matrices. Let's assume that I have an algorithm that is very efficient in transposing 8×8 or 8×16 matrices, but I would like to transpose matrices with an arbitrary size. ...
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0answers
62 views

How do Lie algebra elements act on symmetric and antisymmetric representations?

The Lie algebra of a group acts on itself through the commutator $ T_a \in ad$: $$ T_a \circ T_b = [T_a,T_b] \in ad $$ I assume the same should be true if we have an antisymmetric adjoint, as for ...
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2answers
120 views

If $A$ is an $m\times n$ matrix, $B$ is an $n\times m$ matrix and $n<m$, then $AB$ is not invertible.

The question was given in the early chapters of Linear Algebra by Hoffman & Kunze, so I am trying to give a proof with only the tools given to me so far - which are mainly row reduction and ...
2
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1answer
57 views

Show that $\{w_1,\dots,w_p,v_1,\dots,v_q\}$ is an orthogonal set and spans $\mathbb{R}^n$

These series of questions build up on each other i'm stucked on the last one, i'm also not sure if all of these work but I am pretty convinced they do. Let $W$ be a subspace of $\mathbb{R}^n$ with an ...
5
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1answer
168 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in ...
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3answers
62 views

Dimensions of a basis of a coordinate space

I need a little clarification on the relationship between the basis, its dimension and their corresponding real coordinate space. Suppose we are operating in the fourth coordinate space ...
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2answers
104 views

What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
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23 views

Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
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3answers
43 views

How can I show that and $n\times{n}$ matrix of the form in the description has a determinant of zero for $n>2$?

In General, $n>2$, $a_{i,j}=a_{i,j-1}+1$ and the matrix will be of the following form: ...
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1answer
45 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
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2answers
45 views

Orthonormality of the columns of a matrix

I am studying orthogonal columns and matrices right now and I have encountered the following theorem: Theorem An $m \times n$ matrix $U$ has orthonormal columns if and only if $U^T U = 1$. Is it ...
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1answer
55 views

How to determine positive or negative definite of a bordered Hessian ?

I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully ...