# 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.

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1. ### I [QFT-Schwartz Page. 256] Violation of operator exponentiation rule ?

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...
2. ### I Determinant of a specific, symmetric Toeplitz matrix

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...
3. ### Prob/Stats Books on Combinatorics, Permutations and Probability

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...
4. ### Determining value of r that makes the matrix linearly dependent

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...
5. ### Diagonalizing a matrix given the eigenvalues

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...
6. ### Find the eigenvalues of a 3x3 matrix

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...
7. ### Matrix concept Questions (invertibility, det, linear dependence, span)

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...
8. ### Using a determinant to find the area of the triangle (deriving the formula)

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...
9. ### Finding the Determinant to find out if the matrix is invertible

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...
10. ### Difficult Problem with Matrices

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.🙁🙁
11. ### How to judge the singularity of a matrix in numerical method?

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...
12. ### Simplifying a matrix algebra equation (revised)

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...
13. ### I Beam-splitter transformation matrix

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...
14. ### A Is the Exponential Map Always Surjective from Lie Algebras to Lie Groups?

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...

16. ### Solve simple nonlinear equations in the form [A]x=b

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...
17. ### I Fisher matrix - equivalence or not between sequences

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...
18. ### Find the eigenvalues and eigenvectors

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...
19. ### Proving the following properties

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...
20. ### I Diagonalization and change of basis

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}}\\...
21. M

### Coefficients that make Gaussian elimination impossible?

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...
22. ### Coupled differential equations using matrices

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...
23. D

### Prove trace of matrix: Tr(AB) = Tr(BA)

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...
24. ### Finding the Jordan canonical form of a matrix

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...
25. ### How to Calculate Probability using Density Operator?

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...
26. ### B System of differential equations Basic question

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...
27. ### A What is the closed-form solution using ALS algorithm to optimize

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...
28. ### Understanding the Role of Matrix Multiplication in Solving Equations

Hello hope you can help me. Can anybody tell me what goes on from equation 3 to 4. especially how gets in?
29. ### Matrix-free iteration methods and implicit ODE solvers

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...
30. ### MHB Matrix Algebra 2.0 Help: Solving Questions with Cosine Laws

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...
31. ### Linear Algebra vs Matrix Algebra? Which to pick

In my school LA requires a pre req proof class vs matrix algebra which doesn't. Would matrix algebra even be worth taking?
32. ### Proving a linear algebra equation

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 =...
33. ### How Do You Solve for X in a Matrix Equation?

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)?
34. ### Find matrix representation for rotating/reflecting hexagon

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...
35. ### Proof of (A+B)^2 = A^2 + 2AB + B^2 for Matrix Algebra

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...
36. ### Linear algebra -- compute the following without finding invA

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...
37. ### Tensor Analysis in vector and matrix algebra notation

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
38. ### MHB Matrix Algebra Help: Solve for p1, p2, and p3

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 :)
39. ### How do you reduce a matrix with unknown components?

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...
40. ### Simple Proofs for Matrix Algebra Properties: A Beginner's Guide

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...
41. ### Programs Differential Equations or Matrix Algebra for Physic Major

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...
42. ### Two related questions about matrix algebra

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...
43. ### Matrix algebra - Gauss 0=0, determinant = 0

Homework Statement Mesh Analysis, find current z: A= \left(\begin{array}{ccc}+30x&-15y&-15z\\-15x&+30y&-15z\\-15x&-15y&+30z\end{array}\right) b= \left(\begin{array}{c}+10\\-10\\0\end{array}\right) Homework Equations A*x=b A= resistance x= currents b= voltage sources Gauss elimination...
44. ### Matrix Alalysis, Matrix Algebra, Linear Algebra, what's the difference?

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.
45. ### Engineering Physics, Diff EQ, Matrix Algebra - too hard?

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-...
46. ### How to Calculate T(2, -1, 1) Using Given Linear Transformations?

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...
47. ### Matrix Algebra (Recurrences & Diagonalisation)

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...
48. ### Matrix Algebra Homework: Solving a 6x2 Matrix with Variables to the Second Power

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...
49. ### Help with simple matrix algebra

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...
50. ### Solve X', X for Y: Matrix Algebra Help

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?