What is Matrix algebra: Definition and 65 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. Rlwe

    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...
  2. C

    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...
  3. S

    Determining value of r that makes the matrix linearly dependent

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  4. S

    Diagonalizing a matrix given the eigenvalues

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  5. H

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  6. S

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  7. S

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  8. S

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  9. Physics lover

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  10. N

    How to judge the singularity of a matrix in numerical method?

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  11. T

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  12. R

    I Beam-splitter transformation matrix

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  13. G

    A Lie Algebra and Lie Group

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  14. A

    Matrix algebra : Find the matrix C such that N(A) = R(C)

  15. M

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

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  16. F

    I Fisher matrix - equivalence or not between sequences

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  17. Mutatis

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  18. Abhishek11235

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  19. RicardoMP

    I Diagonalization and change of basis

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  20. M

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  21. Marcus95

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  22. D

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  23. nightingale123

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  24. Guilherme Vieira

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  25. dumbdumNotSmart

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  26. kevin2016

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  27. Martin V.

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  28. P

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  29. S

    MHB Matrix Algebra 2.0 Help: Solving Questions with Cosine Laws

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  30. Lagraaaange

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  31. I

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  32. C

    How Do You Solve for X in a Matrix Equation?

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  33. c3po

    Find matrix representation for rotating/reflecting hexagon

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  34. K

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

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  35. S

    Linear algebra -- compute the following without finding invA

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  36. N

    Tensor Analysis in vector and matrix algebra notation

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  37. M

    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 :)
  38. H

    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...
  39. L

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

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  40. T

    Programs Differential Equations or Matrix Algebra for Physic Major

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  41. K

    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...
  42. D

    Matrix algebra - Gauss 0=0, determinant = 0

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  43. N

    Matrix Alalysis, Matrix Algebra, Linear Algebra, what's the difference?

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  44. H

    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-...
  45. V

    Matrix Algebra Transformation

    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...
  46. S

    Matrix Algebra (Recurrences & Diagonalisation)

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  47. B

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

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  48. R

    Help with simple matrix algebra

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  49. P

    Solve X', X for Y: Matrix Algebra Help

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  50. B

    Matrix Algebra; Linear Dependece; 3 Vectors; In R^4

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