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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3
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0answers
21 views

Prove that $\sum_{i,j} \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0$

Let $v_1 \dots v_n, w_1 \dots w_n \in H$ an inner product space. I am trying (unsuccesfully) to show that $$ \sum_{i,j=1}^n \langle v_i, v_j \rangle \langle w_i, w_j \rangle \geq 0 .$$ Any hints?
2
votes
1answer
68 views

Is every complex number an eigenvalue of some product of three positive definite matrices?

Assume that $A,B$ and $C$ are symmetric positive definite matrices. I guess that the eigenvalues of the matrix $D=ABC$ can be any complex numbers. Is that true?
2
votes
1answer
39 views

Question29 from Contemporary Abstract Algebra [closed]

Consider the element A=(1101) in SL(2,R). What is the order of A? If we view A=(1101) as a member of SL(2,Zp), what is the order of A?
6
votes
2answers
43 views

$AXB$ sort of decomposition?

Let $f: M_n(\mathbb{C}) \to M_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \dots, A_d \in M_n(\mathbb{C})$ and $B_1 \dots, B_d ...
0
votes
0answers
8 views

Relation between Image and kernel of amtrix of sufficient statistics

Suppose we have a parametric statistical model formed by $p_x = \frac{L(\zeta, x)}{\sum_{y\in \mathcal{X}}L(\zeta,x)}$ where $\mathcal{X}$ is the sample space, $L(\zeta,x) = \zeta^{T(x)} = ...
-2
votes
0answers
28 views

What are some unanswered problems on vector algebra? [on hold]

I'm sorry for any mistakes.Math language is different from my country's so they may be so wrong tags. Thanks in advanced ^^
3
votes
0answers
28 views

Matrix multiplication of columns times rows instead of rows times columns

In ordinary matrix multiplication $AB$ where we multiply each column $b_{i}$ by $A$, each resulting column of $AB$ can be viewed as a linear combination of $A$. If however if we decided to multiply ...
-1
votes
2answers
38 views

Prove the matrix $M$ is orthogonal if and only if $M^T= M^{-1}$

Prove the matrix $M$ is orthogonal if and only if $M^T= M^{-1}$ I know that I have to show $$M \text{ is orthogonal } \implies M^T = M^{-1}$$ and $$M^T = M^{-1} \implies M \text{ is orthogonal }$$ ...
0
votes
0answers
7 views

$S(0,\varepsilon ) \Rightarrow F + S(0,\varepsilon ) = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$

Let $F \subseteq {\rm{C}}$ and $S = \left\{ {x \in C:\left\| x \right\| \le \varepsilon } \right\}$. Why does $F + S = \left\{ {\lambda \in C:dis(\lambda ,F) \le \varepsilon } \right\}$? (where ...
1
vote
1answer
20 views

Computationally inexpensive method to find a rotation which minimizes the norm of two tensor difference.

So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ ...
4
votes
1answer
29 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...
-1
votes
1answer
43 views

What value to choose as representative of 100 votes?

I'm trying to use many votes to cast one vote. There are only 4 options, and I'm trying to use "consensus" to decide which to pick. 100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 ...
0
votes
2answers
24 views

Simple exercise regarding space spanned by two vectors

Consider the vectors $X_1=(1,3,2)$ and $X_2=(-2,4,3)$ in $\mathbb R^3$. Show that the set spanned by $X_1$ and $X_2$ is given by $\{(Y_1,Y_2,Y_3):Y_1-7Y_2+10Y_3=0\}$.
3
votes
0answers
27 views

Understanding part of the proof of Spectral Theorem for symmetric matrices

I'm reading a textbook where the Spectral Theorem for symmetric matrices is proven. I understand almost everything about the proof except for one thing. The theorem is stated as follows: Theorem: Let ...
2
votes
2answers
35 views

Linear Algebra - Can vector $v$ be expressed as a linear combination of $u_1$ and $u_2$

I have a question: Can the vector $v = (1,2)$ be expressed as a linear combination of $u_1 = (1,3)$ and $u_2 = (4,1)$? What I have tried: $a + 4b = 1$ $3a + b = 2$ $a = 1 - 4b$ $3(1 - 4b) +b = 2$ ...
1
vote
1answer
31 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2\cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=\ ?$ Is ...
0
votes
0answers
37 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
1
vote
1answer
26 views

Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ being solved?

Let A be a positive semidefinite matrix of order $n$. Is the conjecture $Per A $ is the largest eigenvalue of $\tilde{A}$ is being solved? Where $\tilde{A}$ is the matrix of order $n!\times ...
2
votes
1answer
30 views

Prove that this matrix is total unimodular

Is there an easy way to prove that this matrix is total unimodular ? $$ \begin{bmatrix} 1 & F_1 & 0\\ 1 & 0 & F^T_1 \\ 0 & F_2 \end{bmatrix} $$ $1$ is the identity matrix, ...
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votes
2answers
73 views

What is the reduced row echelon form of $A$? [on hold]

Let $$A = \left( \begin{array}{cccc} 7 & 7 & 9 & -17\\ 6 & 6 & 1 & -2 \\ -12 & -12 & -27 & 1 \\ 7& 7 & 17 & -15\end{array} \right)$$ What is the ...
0
votes
0answers
13 views

steps involved in matrix algebra problem

If $E$ is a column vector, $\Sigma$ is an $n \times n$ symmetric matrix, Let : $$ A = E^T \Sigma^{-1}E \quad~~~~~~ B = E^T \Sigma^{-1}1 ~~~~~\quad C= 1^T \Sigma^{-1}1 \quad $$ Then let: $$ w = ...
1
vote
1answer
10 views

Compactness of a linear operator

The question is as follows: Show that a linear operator $T:X\to Y$ where $X$ and $Y$ are normed spaces is compact if and only if the image $T(M)$ of the unit ball $M\subset X$ is relatively compact ...
3
votes
0answers
41 views

Does this binary operation have a name?

Does the following binary operator on abstract linear maps $A,B:\mathbb{C}^N \rightarrow \mathbb{C}^N$, have a name: $[\{A,B\}]:= AB^{\dagger} - BA^{\dagger}$ clearly, it is real bi-linear, but not ...
0
votes
0answers
56 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000...}_{n-1 times} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 = ...
1
vote
1answer
19 views

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P ...
0
votes
0answers
16 views

vector 3d rotation of a cube

I have a cube which is rotated by plane you can see it in an example here. What am I trying to achieve is algorithm that tells what is the top, face and side after a rotation is performed. And also ...
6
votes
1answer
23 views

Verification for a block-determinant evaluation, and some further thoughts

First, I want some verification for the validity of my approach for this det evaluation question: If $A,B\in M_n(K)$, $K$ is a number field (in the sense that $\Bbb Q$ is the smallest possible ...
8
votes
1answer
87 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like fourier transform - its actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
11
votes
2answers
383 views

Any neat way to calculate this Vandermonde-like determinant?

Let $x_i,i\in\{1,\cdots,n\}$ be real numbers, and $s_k=x_1^k+\cdots+x_n^k$, I'm asked to calculate $$ |S|:= \begin{vmatrix} s_0 & s_1 & s_2 & \cdots & s_{n-1}\\ s_1 ...
0
votes
1answer
32 views

A program to visualize Linear Algebra?

I am asking here because I believe you have some idea of a good visualizer 3d program to see what are really: eigenvectors, subspaces, rowspaces, columnspaces and just answers on normal matrix ...
0
votes
0answers
42 views

Matrix of a differential equation

I had recentely encounter my first exercise about merging matrices notions and differential equations functions, but after solving the differential equation, I don't know how to represent it in the ...
0
votes
0answers
10 views

Some “facts” on oriented angles in the Euclidean affine space of dimension 2

Let $\mathcal{E}^2$ be an Euclidean affine space of dimension $2$ oriented by $R=(O,B=\{u_1,u_2\}$. Let $$\mathcal{r}=\{P \in \mathcal{E}_2 \text{ such that } \overrightarrow{AP} \in \langle u ...
4
votes
1answer
69 views

“Hard” exercises on Linear Algebra and Analytic Geometry

I started lecturing this subject called "Linear Algebra and Analytic Geometry" and in the second day of class I was approached by an undergrad student, asking for referenced that would contain "hard" ...
0
votes
0answers
15 views

Linear dependence of rank one approximations.

In my research, I am faced with the following problem. Let $\mathbf{X}$ and $\mathbf{Y}$ be two nonzero $m \times n$ matrices with entries in some field $\mathbb{K}$. Let $\phi:\mathbb{K}^{m\times ...
0
votes
1answer
15 views

Matrix operation: putting the rows next to each other

I have a matrix $A$ of dimension $N\times K$, and want to find a way to convert it to a matrix $B$ of dimension $1\times NK$. For example: ...
-4
votes
2answers
41 views

Using the harmonic mean to determine the time to perform a task with varying manpower. A/K/A “Frank, Bob and Diane” [on hold]

Each year West Coast Shipping provides transportation to Pebble Beach Concours D'elegance. Our 3 drives Frank, Bob and Diane can deliver all the vehicles from the auction to West Coast Shipping ...
3
votes
3answers
58 views

Inner Product, Orthogonality, and Coordinate Systems

I remember my professor saying there are certain advantages to using an orthogonal basis. One is that it's easy to determine the coordinates of a given vector. For example, we are familiar with the ...
1
vote
2answers
44 views

Balancing chemical equations using linear algebraic methods

I know there are already plenty of questions on this site regarding this topic but I am having difficulty with a particular chemical equation. I am trying to balance the following: $$ { C }_{ 2 }{ H ...
0
votes
2answers
35 views

I'm having a problem with this number and digits problem. What to do?

During the last election , the total number of votes recorded in the municipality of San Juan was 8600. Had one-third of Estrada’s supporters stayed away from the polls and one-half of Arroyo’s ...
0
votes
2answers
442 views

Rank of a Matrix and Echelon Form to determine ranks.

What is the meaning rank of a matrix in terms of vectors, and how does Echelon form work in determining the rank of a matrix?
1
vote
1answer
44 views

Linear Systems: Exponentials of a Matrix

I have a rather odd question to some, but one that has stumped me for a good few minutes on a homework assignment that states: For each matrix, find the eigenvalues of $\text{exp}{(A)}$, ...
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votes
0answers
34 views

Linear Algebra Subspace question, set difference [on hold]

If $W_1$ and $W_2$ are subspaces of a vector space $V$, is $W_1 \setminus W_2$ ever a subspace of $V$? Why or why not? (Here $W_1 \setminus W_2$ denotes the set difference of $W_1$ and $W_2$: $W_1 ...
0
votes
0answers
19 views

How can we multiply partitioned matrices?

Im trying to perform the following product: Question: can we just think of each block as an element and perform the multiplication like matrix multiplication?
3
votes
0answers
47 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
2
votes
2answers
38 views

How did they derive the image from kernel?

I understand its something to do with the rank nullity theorem, but im not sure how they applied it to get the basis of the image. By my understanding, they took the leading entries of the rows of ...
1
vote
1answer
23 views

Linear equation and linear differential equations

I remember noting from an algebra class that $x$ and $y$ of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations: General form of ...
2
votes
2answers
43 views

What is the matrix $\left[ DS(A) \right]$, which gives $\left[ DS(A) \right] H=AH+HA$?

In Hubbard's multivariable calculus book $DS(A):H \mapsto AH+HA$ is introduced as a linear transformation where $A$ is an $n \times n$ matrix, $S(A)=A^2$, and $D$ is the notation for derivative. It ...
0
votes
1answer
18 views

Linear Systems and Linear Transformation

I want to confirm my attempt to see if I am on the right track. The question is as follows. Show that the operator norm of a inear transformation $T$ on $\mathbb{R}^n$ satisfies ...
-1
votes
1answer
25 views

Understanding Composition Function (fg)(-1) for f(x)=x-3 & g(x)=x^2-8x+15?

Can someone help explain how to do the following composition function to me? (or at least get me started) Find the value of (fg)(-1) if ...
0
votes
2answers
35 views

Is the maximum of the eigenvalues of any symmetric positive?

Let $A$ be a symmetric matrix having dimension $n \times n \;, \; \; n\geq 2$. If one wants to pick the maximum of its eigenvalues, will the value be positive? Suppose A was an adjacency matrix, ...