For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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3
votes
1answer
28 views

Finite series for the matrix exponential

It is well know that any analytic function of an $n \times n$ real/complex matrix $f(A)$ can be expressed as linear combination of the first $n$ powers of $A$ by the Cayley-Hamilton theorem. Is it ...
1
vote
2answers
25 views

Convert a linear system to higher order 1D equation

There are numerous explanations of converting an $n^\text{th}$ order differential equation $$\sum_{i=0}^n a_i \frac{\partial^i x}{\partial t^i} = 0 $$ to a linear system with $n$ dimensions. ...
0
votes
0answers
8 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 ...
0
votes
1answer
24 views

Fibonacci sequence in system of equations?

Can we write/solve the fibonacci sequence in a linear system of equations, for a given number of terms? I know we can define the recursive definition using matrices but what i am interested in is ...
1
vote
1answer
21 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 = ...
3
votes
0answers
29 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 ...
0
votes
0answers
26 views

Using mathematics how it is possible to define a model or formula which validates “kinetic movements of objects” in 3-dimensional space? [on hold]

I'm considering a matrix of MxN dimensions its shown as grid in shown diagram below. These grid are "objects" in physical world used to define certain state of affairs and have existential properties ...
-1
votes
2answers
41 views

By using the properties of determinant show that

$$\begin{vmatrix}1&a&a^2\\ 1&b&b^2\\ 1&c&c^2\end{vmatrix}=(b-a)(c-a)\begin{vmatrix}1&a&0\\ 0&1&b\\ 0&1&c\end{vmatrix}$$ I have been trying to solve ...
6
votes
1answer
25 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 ...
1
vote
1answer
24 views

Problem understanding how a linear equation is simplified

Using this paper as a reference (Section IV.C, page 4318), We have the following objective function which we wish to minimize with respect to $D \in \mathbb R^{n \times K}$ ($X \in \mathbb R^{K \times ...
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 ...
-4
votes
1answer
15 views

Norm of a matrix that has integral operator as its entries? [on hold]

What is the norm of a matrix that has integral operators as its entries? For example A_{11} A_{12} A_{21} A_{22} where $A_{ij}$ are integral ...
0
votes
1answer
16 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: ...
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
0answers
43 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
1answer
15 views

Easiest Way to Find The Inverse Matrix by using Row Canonical Form.

I want to find the Inverse of a Matrix in Row Canonical Form . I have tried several ways to do that but failed. Is there any easy way to find the Inverse of Matrix in row canonical form? Let A be 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
52 views

incidence matrix of a digraph with a self loop

When writing the incidence matrix of a digraph, we denote '+1' if the tail of an arc is at the vertex, '-1' if the head is at the vertex. Is it '0' for a self loop at the vertex?
0
votes
1answer
54 views

Does $A^2$ similar to $B^2$ imply that $A$ is similar to $B$? [duplicate]

So we have square matrices $A$ and $B$. Now suppose $A^2$ and $B^2$ are similar, does it follow that $A$ and $B$ are similar? I don't think so, but I'm having trouble showing is not. My attempt: If ...
3
votes
1answer
39 views

Are $B=PAP^{-1}$ and $B=P^{-1}AP$ equivalent?

Im looking at the solution to one of my questions. Basically, we started off with a matrix $A$ (in the elementary basis) which we want to convert into a diagonal matrix $B$ of another basis. Question ...
0
votes
0answers
39 views

Finding the eigenvalues of this matrix.

Suppose I wanted to find the eigenvalues of the two by two matrix A: $$ A=\pmatrix{i&1\\0&1} $$ We see that $A(x,y)=(xi+y,y)$ I see that $\lambda = i$ is an eigen value, with the ...
-2
votes
2answers
30 views

Rank of an $m$ by $n$ matrix?

Can anyone state, in plain English, how to find the rank of an $m$ by $n$ matrix? Is it necessary to perform Gaussian elimination first, or translate it into upper triangle form (or however it is ...
0
votes
1answer
20 views

Prove the following using induction on d (matrices)

I manage to reach the step where I need to prove n = k + 1 but I am battling to complete the proof as I am not certain what to do with the exponents in my answer. I will run through the proof as I ...
0
votes
1answer
18 views

Count of solutions to matrix equations

Given these modular equations: $$a_{1,1} x_1 + a_{1,2} x_2 + \cdots + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + \cdots + a_{2,n} x_n = b_2 \bmod p $$ $$\vdots$$ $$a_{m,1} x_1 + ...
7
votes
4answers
512 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
5
votes
1answer
79 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...
1
vote
3answers
25 views

Checking whether the result is positive definite or positive semi-definite with two methods

Given, $$A = \begin{bmatrix} 1 &1 & 1\\ 1&1 & 1\\ 1& 1& 1 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Using Method 1: ...
2
votes
2answers
46 views

What is the good way to remember the signs of the rotational matrix?

Recall rotational matrix in (x,y) is given by: $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ For the life of me I cannot remember if the ...
2
votes
4answers
38 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
-5
votes
0answers
31 views

Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
0
votes
1answer
26 views

Nullspace of Original Matrix multiplied by Transpose

I need help in a question. I need to show that, given an $ m \times n$ matrix $A$, $\bar x \in Null(A)$ if and only if $\bar x \in Null(A^tA)$. I found this answer: "Let $X=Null(A)$; then $\forall x ...
0
votes
1answer
32 views

If I have $m \times n$ matrix A and a vector $x \in \mathbb{R}^m$, Can I make Ax working? Will it be possible for Ax to do row operations?

If I have $m \times n$ matrix A and a vector $x \in \mathbb{R}^m$,where m>n, Can I make Ax working? Will it be possible for Ax to do row operations? If yes, then how do they operate?
-1
votes
1answer
18 views

Making a basis from the Column Space of a Matrix in MatLab [on hold]

Starting with matrix A whose entries are all zeros or ones, I want to make a new matrix B whose columns form a basis for the column space of A. I know that rref puts A in Gauss Jordan form and the ...
0
votes
0answers
47 views

How many $2\times3$ real matrices are needed to guarantee that at least one of them is a linear combinations of the others?

The only thing I know is that $$\left(\begin{array}{ccc}1&0&1\\0&1&1\end{array}\right)$$ Seems to have a column to be linear combinations of the others.
0
votes
1answer
20 views

Deducing bounds on operator norms of matrix differences

Let $A$ and $B$ be two $n\times n$ matrices with entries $0\leq A_{ij}, B_{ij}\leq 1,$ for all $i,j.$ Suppose we are given a bound on the operator norm as follows: $$\|A-B\|\leq \delta.$$ What is a ...
0
votes
0answers
21 views

Finding the maximum Eigen vector in MATLAB

I am trying to apply something I learned in paper regarding finding the maximum eigenvector of a matrix. Lets assume the matrix is $T$ and the max eigen vector of matrix ${\bf T}$ is ${\bf w}$ $${\bf ...
0
votes
1answer
23 views

How to calculate the differential of vector/matrix?

Suppose we have $L=AXB$, where $A\in R^{m\times n}$, $X\in R^{n\times p}$ and $B\in R^{p\times q}$. Then, how can we obtain the following differential: $$\frac{\partial L}{\partial X}.$$ If possible, ...
0
votes
0answers
19 views

Finding a variable in the determinant of sum of matrices [on hold]

I Don't Know how to earn P that is scalar from Below Formula : R = log2(abs(det(I + P * H*H'))) Everything is known except P. P is scalar and positive. I is an Identity NxN , H is complex NxN ...
2
votes
1answer
27 views

If I had a matrix A, what is the meaning of $A^T Ax$, given $A^T$ is transpose of A and x are vectors of variable?

If I had a matrix $A$, what is the meaning of $A^TAx$, given $A^T$ is the transpose of $A$ and $x$ is a vector? Is it operation on $x$ by the result of the multiplication of two matrices, or is it ...
4
votes
4answers
80 views

Matrix exponential: $\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$

It is asked to calculate $e^A$, where $$A=\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$$ I begin evaluating some powers of A: $A^0= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\; ; ...
0
votes
1answer
24 views

What should I do to tackle the following matrices calculation?

Through chapter 3 of Group Theory by Morton Hamermesh in part 3-6 (Equivalent representations; characters.) I stopped in some point. It's told "If we change the basis in the n-dimensional space $L$, ...
0
votes
1answer
21 views

Confusion between eigen value decomposition and singular value decomposition

The Singular Value Decomposition of matrix $H$ gives $$H = U \Sigma V^H$$ The Eigen value decomposition of $$HH^H= U \Sigma \Sigma^t U^H$$ I took an example in matlab and performed EID and SVD ...
1
vote
0answers
19 views

Calculating central elements of Universal Enveloping Algebras?

Simply put, how do I calculate (in general) the central elements of the UEA of some Lie algebra given some desired degree in the algebra generators? I know the so-called 'quadratic Casimir', of ...
1
vote
1answer
28 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$\text{Im} \phi^m=\text{Im} ...
3
votes
1answer
54 views

Show that $Ax=0, Bx=0$ share the same solution space iff there is some invertible $P$ s.t. $B=PA$.

The question is said in the title, suppose $A,B\in M_{m\times n}(K)$, where $K$ is some infinite number field. If we regard $A,B$ as linear maps from $K^n$ to $K^m$, then they share the same ...
2
votes
1answer
69 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?
0
votes
1answer
29 views

Matrix polynomials/eigenvalues

$\begin{pmatrix} 7 & -2\\2 & 2 \end{pmatrix}$ The eigenvalues for this matrix are $\lambda=6$ and $\lambda=3$ It also happens that $(A-6I)(A-3I)=0$ I've checked for various $2$ x $2$ ...
-2
votes
0answers
22 views

A question on matrix norm

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
-1
votes
1answer
45 views

Three lines that intersect in a plane.

Find a condition for three lines (𝑖 = 1,2,3) in a plane given by $π‘Ž_𝑖 π‘₯ + 𝑏_𝑖 𝑦 = 𝑐_𝑖$ to intersect in one point. I decided to form a matrix and to find the identity matrix since it will ...
1
vote
0answers
16 views

Property of hermitian matrices (eigen values)

In a paper I have read the following ${\bf G}$ is a Hermitian matrix, that 1) ${\bf G}$ is diagonalizable 2) the singluar values are same as the eigen values Is number 2 correct? I cant seem to ...