What is Matrix multiplication: Definition and 107 Discussions

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in 1812, to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering.
Computing matrix products is a central operation in all computational applications of linear algebra.

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  1. M

    Is result of vector inner product retained after matrix multiplication?

    Hi, I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea. Question: Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve...
  2. L

    A Matrix multiplication, Orthogonal matrix, Independent parameters

    Matrix multiplication is defined by \sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw \sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij} This is because ##A^TA=I##. How to know how many independent parameters we have in...
  3. M

    MATLAB Matrix multiplication without a for-loop for an uneven size matrix and a vector

    Hi PF! I am trying to multiply each component of B by the matrix A and then solve A\C. See the code below. A = rand(4); B = rand(5,1); C = rand(4,1); for i = 1:5 sol(:,i) = (B(i)*A)\C end But there has to be a way to do this without a for-loop, right? I'd really appreciate any help you have!
  4. J

    Rotation by matrix multiplication -- confirmation please

    The below matrix represents a rotation. 0 0 -1 0 1 0 1 0 0 Im trying to obtain the general point (x y z) when rotated by the above rotation matrix? So visited https://www.andre-gaschler.com/rotationconverter/ entered the above figures and not sure which entry would be x y z but assume it...
  5. Athenian

    I Lorentz Boosts: Finding Speed, Coordinates & Rotation w/ Matrix Multiply

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  6. C

    Matrix Multiplication -- Commutivity versus Associativity

    According to me matrix multiplication is not commutative. Therefore A^2.A^3=A^3.A^2 should be false. But at the same time matrix multiplication is associative so we can take whatever no. of A's we want to multiply i.e A^5=A.A^4 OR A^5=A^2.A^3
  7. T

    Matrix Multiplication Homework: Equations and Solutions"

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  8. B

    B Associativity of Matrix multiplication

    ##\begin{align}[A(BC)]_{ij} &= \sum_r A_{ir}(BC)_{rj} \\ &= \sum_r A_{ir} \sum_s B_{rs}C_{sj}\\ &= \sum_r\sum_s A_{ir}B_{rs}C_{sj}\\ &= \sum_{s} (\sum_{r} A_{ir} B_{rs}) C_{sj} \\ &= [(AB) C]_{ij}\end{align}## How did it went from ##2## to ##3##. In general is there a proof that sums can be...
  9. R

    I Why is the dot product equivalent to matrix multiplication?

    Why is the dot product equivalent to the matrix multiplication of its components? I've seen some proofs using Pythagorean and cosine law but they don't give you an intuitive feel as to why matrix multiplication works. The geometric definition (##ab cosθ##) is very easy to understand. To a...
  10. M

    Show GL/O/SO(n,R) form groups under Matrix Multiplication

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  11. MickeyBlue

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  12. O

    I What does adjacent indices mean in the context of matrix multiplication?

    Hello, I was refreshing my Mathematics using S.M. Blinder's book "Guide to Essential Math" and on the section on Matrix Multiplication I got the following, Can someone elaborate on the highlighted section? In particular, what does "adjacent indices" mean? Thank you.
  13. M

    Software for multiplication of matrices

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  14. TheSodesa

    Proving two simple matrix product properties

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  15. P

    MHB Sava's question via email about matrix multiplication

    $\displaystyle \begin{align*} A\,A^T &= \left[\begin{matrix} 3 & 0 & -4 \\ 4 & 0 & \phantom{-}3 \\ 0 & 5 & \phantom{-}0 \end{matrix}\right]\left[ \begin{matrix} \phantom{-}3 & 4 & 0 \\ \phantom{-}0 & 0 & 5 \\ -4 & 3 & 0 \end{matrix}\right] \\ &= \left[ \begin{matrix} 3\cdot 3 + 0 \cdot 0 +...
  16. D

    Linear Algebra Book about block matrix multiplication

    I still can't find a book with properties and theorems involving block matrices multiplication to reference in my undergraduate work. thanks
  17. D

    Master Matrix Multiplication: Solving Size Confusion | Homework Help

    Homework Statement Well, basically my issue isn't exactly with how to multiply matrices. I know how to do that, my issue is the way my lecturer shows how to find the size of the new matrix, this is all that he shows: I mean how is he getting AX to be a 3x1 matrix? Homework EquationsThe...
  18. N

    Matrix multiplication, P1.1.1 Golub/Van Loan-Matrix Computations 3rd

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  19. C

    Matrix multiplication - is this plausible?

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  20. C

    MHB Matrix multiplication simplified to Vector multiplication

    Hello, I'm not sure where to put this. I have spent the last week (14+ hour days) editing some code I have for selecting representative spectra for a remote sensing masters thesis I'm working on. The program is very-very slow, and I've been trying to speed it up as much as possible by NOT...
  21. R

    A question about the cross product as related to matrix multiplication

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  22. PsychonautQQ

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  23. ajayguhan

    Matrix Multiplication: Finding A^50 with a Shortcut Method

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  24. X

    Explain why matrix multiplication is not commutative.

    The title says it all. Commutative* sorry Mod note: fixed title.
  25. ajayguhan

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  26. G

    MHB Problem involving matrix multiplication and dot product in one proof

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  27. AJBMuir

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  28. S

    Matrix Multiplication and Function Composition

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  29. Y

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  30. Nugso

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  31. E

    Divide and Conquer, Matrix Multiplication MATLAB

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  32. D

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  33. A

    Using Matrix Multiplication to Obtain Another Matrix

    Homework Statement I need to use matrix multiplication of matrices A-D to obtain matrix E. I also need to set a equal to some value that would allow me to perform this multiplication. Homework Equations The matrixes I need to multiply: A = [1, 1; 0, 1] B = [1, 0; 1, 1] C = [a, 0...
  34. Y

    MHB Property of Matrix Multiplication

    Hello, I wanted to ask if this is a correct move, A and B are matrices, a is a scalar, thank you ! A^{2}\cdot B^{t}-aA=A(A\cdot B^{t}-aI)
  35. S

    Help with cblas_dgemm matrix multiplication

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  36. Y

    MHB Matrix Multiplication: Is A Solution Possible?

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  37. A

    Continuity of matrix multiplication and inversion in a normed vector space?

    Homework Statement Hi guys, I'm trying to prove that matrix inversion is continuous. In other words, I'm trying to show that in a normed vector space the map \varphi: GL(n,R) \to GL(n,R) defined by \varphi(A) = A^{-1} is continuous.Homework Equations The norm that we're working in the...
  38. B

    Matrix Multiplication and sigma notation

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  39. A

    Why do we use double pipes to represent the norm of a vector?

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  40. R

    Proof Involving Matrix Polynomials and Matrix Multiplication

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  41. haael

    Cross product and matrix multiplication

    Suppose that we have a cross-product of vectors. a × b = c Now suppose that we have an orthogonal matrix M. Is it true that (M a) × (M b) = M c ? My intuition is that here we are moving to another coordinate system and performing a cross product in this new system. I can't find an...
  42. W

    Euler transform matrix multiplication help

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  43. J

    Matrix multiplication vs dot product

    What is the difference between matrix multiplication and the dot product of two matrices? Is there a difference? If, A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} e & f \\ g & h \end{pmatrix} then does {\mathbf{A} \cdot...
  44. H

    Matrix Multiplication or Inverse Problem

    Homework Statement I am more trying to figure out how to solve generally rather than solve this specific problem. Nevertheless this problem could be given as: Solve the matrix for A and B. Homework Equations \begin{pmatrix} 1 & 1 \\ 0 & 0 \\ 0 & 2 \\ 2 & 0 \\ 0 & 0 \\...
  45. P

    Matrix Multiplication to translate object with given vertices

    This was a question for a test of mine. I am unsure how to translate the object from the left image(Fig.1) to the right (Fig.2). I am to use matrix multiplication.. Do i start with the vertices in Fig.1 as a matrix, as in |0 -1 -2 0 -2 -1 | ...............|0 0 0 2 2 3 | (I...
  46. S

    System under matrix addition (+) and matrix multiplication (.) is a field

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  47. homology

    Raising/Lowering indices and matrix multiplication

    Please read the following carefully. The point of the following is to distinguish between T^{\mu}_{\mbox{ } \nu} and T_{\mbox{ }\mu}^{\nu} which clearly involves a metric tensor. But when you want to go from component manipulation to matrix operations you have to be careful. Components are...
  48. M

    Three-Dimensional Matrix Multiplication

    Does this concept exist? Google yields weird results that mostly have to do with programming, and Wikipedia says nothing about it. I always find that I understand tensor math better when I can translate it into matrix notation, but if I'm dealing with tensors of too high a rank, I don't know...
  49. P

    Matrix Multiplication: Calculating (AB)C and A(BC) Using the Formula

    A is an M × N matrix, B is N × K and C is a K × L matrix. Consider matrix multiplication (AB)ij = Pk AikBkj . Using the formula, (AB)ij = Pk AikBkj, how would I calculate ((AB)C) and (A(BC))?
  50. R

    Matrix Multiplication and Rank of Matrix

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