What is Matrix algebra: Definition and 66 Discussions
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication (Lam 1999). The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs.
When R is a commutative ring, the matrix ring Mn(R) is an associative algebra over R, and may be called a matrix algebra. In this setting, if M is a matrix and r is in R, then the matrix rM is the matrix M with each of its entries multiplied by r.
When computing the projection of time-evoluted state ## |x_j> ## on ## |x_{j+1}> ## it uses the 'completeness' of momentum basis ## \int \frac{dp}{2\pi} |p><p| ##. Next it explicitly states the form of Hamiltonian ## \hat{H} = \frac{\hat{p}^2}{2m}+\hat{V}(\hat{x_j},t_j) ##. Thereafter i believe...
Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...
Hello!
I am looking for textbooks to relearn Combinatorics, Permutations Combinations and Probability and also Matrix algebra( decomposition, etc). I had done these many years ago and the course/books provided to me at that time weren't that great. So I want to relearn this with a more...
for problem (a), all real numbers of value r will make the system linearly independent, as the system contains more vectors than entry simply by insepection.
As for problem (b), no value of r can make the system linearly dependent by insepection. I tried reducing the matrix into reduced echelon...
The following matrix is given.
Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1.
The answer sheet reads that the diagonal matrix D is as follows:
I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...
Hi,
I have a 3 mass system. ##M \neq m##
I found the forces and I get the following matrix.
I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values.
I'm wondering if...
I have a trouble showing proofs for matrix problems. I would like to know how
A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix
holds for square matrix A. It would be great if you can show how one leads to another with examples! :)
Thanks for helping...
This is the question. The following is the solutions I found:
I understand that the first line was derived by setting one vertex on origin and taking the transpose of the matrix. However, I cannot understand where the extra row and column came from in the second line. Can anyone explain how...
question:
My first attempt:
my second attempt:
So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...
I assumed a column vector of degree 3 and then calculated A from the given condition.But after solving it i tried to find A2 and then I got stuck as none of the options seem to match.
Please help.
I think i will have to learn LATEX.🙁🙁
Summary: different methods give different results. They are not consistent.
Summary: different methods give different results. They are not consistent.
I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...
I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...
The transformation matrix for a beam splitter relates the four E-fields involved as follows:
$$
\left(\begin{array}{c}
E_{1}\\
E_{2}
\end{array}\right)=\left(\begin{array}{cc}
T & R\\
R & T
\end{array}\right)\left(\begin{array}{c}
E_{3}\\
E_{4}
\end{array}\right)
\tag{1}$$
Here, the amplitude...
Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...
Hi! I have a simple set of nonlinear equations
1) 3x = 30
2) x+2y = 20
3) x + y*z = 15
Clearly the solution to this is (10,5,1) but I want to find a robust way to solve this type of problem [A]x=b (where [A] is a simple function of x) which doesn't involve numerically solving using Newtons...
I am currently studying Fisher's formalism as part of parameter estimation.
From this documentation :
They that Fisher matrix is the inverse matrix of the covariance matrix. Initially, one builds a matrix "full" that takes into account all the parameters.
1) Projection : We can then do...
Homework Statement
Find the eigenvalues and eigenvectors fro the matrix: $$
A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$.
Homework Equations
Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## .
The Attempt at a Solution
I've found...
Mentor note: Member warned that an attempt must be shown.
1. Homework Statement
This question is from book Afken Weber, Mathematics for Physicist.
An operator ##T(t + ε,t)## describes the change in the wave function from t to t + ##\epsilon## . For ##\epsilon## real and small enough so that...
I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle:
\begin{bmatrix}
E_0 &-A \\
-A & E_0
\end{bmatrix}
Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix}
\frac{1}{\sqrt{2}}\\...
Homework Statement
Given this matrix:
I am asked to find values of the coefficient of the second value of the third row that would make it impossible to proceed and make elimination break down.
Homework Equations
Gaussian elimination methods I used given here...
Homework Statement
We can treat the following coupled system of differential equations as an eigenvalue
problem:
## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ##
## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ##
## \frac{dy_3}{dt} = f_3 - 4y_3 ##
where f1, f2 and f3 is a set of time-dependent sources, and...
Homework Statement
[/B]
The trace of a matrix is defined to be the sum of its diaganol matrix elements.
1. Show that Tr(ΩΛ) = Tr(ΩΛ)
2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic)
my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U...
Homework Statement
About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know.
$$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$
Find the
a) Jordan canonical form of ##A##
b) characteristic polynomial
c) minimal polynomial
d) ##dim\,kerA##
When:
case 1: we know that ##A## is...
Hello, I'm trying to understand how to calculate de probability of finding a system in a specific eigenstate using the density operator. In the book of Balian, Haar, Gregg I've found a good definition of it being the expectation value of the projector Pr in the orientation of the eingenstate...
So I ran into an case I have not seen before. Say we have a system of 3 equations such that W´=AW, where W=(x(t),y(t),z(t)) and A is a 3x3 matrix. The way I usually approach these is by finding the eigenvalues of A to then find the eigenvectors and thus find the ¨homogenous¨ solution. What...
C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n}
##{S}## and ##{P}## are similarity matrices (symmetric).
##\lambda##, ##\alpha## and ##\beta## are...
Im trying to implement the implicit Euler method in high-performance software for micromagnetic simulations, where I'm restricted to using the driving function of the ODE (Landau-Lifshitz equation) and the previous solution points. This obviously not a problem for an explicit method, since we...
Hey guys,
So I'm stuck on another question from the previous one that I posted and would absolutely love it if I can get some help regarding how to attempt this. I literally have no clue at how to go by solving it. I have a feeling for question one that the cosine laws might come in handy but...
I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent.
I would appreciate any help on this.
Prove A = B
A = (Σπ^-1 + Σy^-1)^-1 * (Σπ^-1*π + Σy^-1*y)
y = π+ X*β
Σπ =τ*Σ
Σy = X' * Σβ * X + ΣεB =...
Homework Statement
Given the matrices A, B, C, D, X are invertible such that
(AX+BD)C=CA
Find an expression for X.
Homework Equations
N/A
Answer is A^{-1}CAC^{-1}-A^{-1}BD
The Attempt at a Solution
I know you can't do normal algebra for matrices.
So this means A≠(AX+BD)?
Homework Statement
Consider the set of operations in the plane that includes rotations by an angle about the origin and reflections about an axis through the origin. Find a matrix representation in terms of 2x2 matrices of the group of transformations (rotations plus reflections) that leaves...
This problem is so simple that I'm not exactly sure what they want you to do:
Let A and B be n x n matrices such that AB = BA. Show that (A + B)^2 = A^2 + 2AB + B^2. Conclude that (I + A)^2 = I + 2A + A^2.
We don't need to list properties or anything, just manipulate. This all seems...
Homework Statement
Homework Equations
A=LU, U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2,
The Attempt at a Solution
I used MATLAB and the relations:
U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2,
to find a solution
I found U^-1*L^-1 , let =B...
Is there anywhere that teaches tensor analysis in both tensor and non tensor notation, because I'm having to pause each time i look at something in tensor notation and phrase it mentally in non tensor notation at which point it becomes staggeringly simpler. Any help apreciated
Hopefully someone can help me solve this, I'm usually quite good at matrix algebra but for some reason I cannot solve this equation.
p1+4p2+8p3=26
5p1+7p2=38
8P1+12p2+2p3=66
If somebody could help me with the values of p1 p2 p3 that would be a great help :)
Hi, I've been running into a problem lately where I have a system of equations that needs to be solved or I need to do some other sort of matrix algebra, but the components of the matrix that I am trying to perform row operations on have unknowns in them.
Specifically, I was working with a...
Hello,
So I am struggling with a couple very simple proofs of properties of matrix algebra. This is the first time I have ever had real proofs in math (Linear algebra). For the first one, I have it from our text but need a little help, and I am completely lost on the second one.
1) Prove...
I am signing up for my third quarter of college classes soon and I have to choose if I am going to take Differential Equations or Matrix Algebra this quarter. I am given the option to take either of them and I do not know if I will be able to take the other one anytime soon. Which of the two...
Homework Statement
a.) If A is an 'n x n' matrix and X is an 'n x 1' nonzero column matrix with
AX = 0
show, by assuming the contrary, that det(A) = 0
b.) Using the answer in 'a' show that the scalar equation which gives the values of λ that satisfy the matrix equation AX = λIX is:
det(A...
Matrix Alalysis, Matrix Algebra, Linear Algebra, they seem to cover many similar topics.
Would someone explain about what are the differences between them?
Thanks in advance.
Hello, I am currently signed up for Fall Quarter Engineering Physics, Diff EQ, and Matrix algebra at the University of Washington.
Right now I am sort of skeptical if I should drop one of these classes because I have heard horror stories from at least one of each of these classes. I got a B-...
Homework Statement
Let T: R3 → R3 be a linear transformation such that T(1, 1, 1) = (1, 0, –1), T(0, – 1, 2) = (–3, 3, –1), and T(1, 0, 1) = (1, 1, 0). Find the following expression. (Enter each vector as a comma-separated list of its components.)
what is T(2, –1, 1)?
The Attempt at a...
Solve this simultaneous pair of recurrences using diagonalisation
Not sure what would be related equations to this.
Originally I had no idea how to do this, I set up the first matrix, like this.
Then, from there, I know that I have to let:
x_k= [[c_k][d_k]]. Then x_k...
Homework Statement
For the last 90 minutes I've been working on this problem.
https://www.physicsforums.com/showthread.php?t=582722
I don't know if it's been solved but I don't care because I really like the challenge of it. I'm at the point now in the problem where my knowledge of matrix...
Hi all,
I'm having trouble solving this matrix problem, basically I have,
s=r*H
where s = [ 1 1 0] and,
H=
1 0 1 1 0 0
0 1 1 0 1 0
1 1 0 0 0 1
I am trying to find out what the matrix r is but it won't work in matlab. I have tried...
Find X if
X' * X =
(50, 300)
(300, 1000)
Then find Y if
Y * X =
(300, 2000)
Problem is that when I try to solve for X (or X') first, I get three equations with four unknowns (a11, a12, a21, a22). Any help?