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

    Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening: $$ \left( \begin{array}{cccc} \frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\ 0 & 1 & 0 & 0...
  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"

    Homework Statement Homework Equations Matrix multiplication. The Attempt at a Solution Answer given=4 What am I doing wrong??
  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

    Homework Statement Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). Homework EquationsThe Attempt at a Solution So I know the properties that define a group are...
  11. MickeyBlue

    Representing a transformation with a matrix

    Homework Statement Use matrix multiplication to find the 2×2 matrix P which represents projection onto the line y =√3x. Can you suggest another way of finding this matrix? Which vectors x∈R2 satisfy the equation Px = x? For which x is Px = 0? Homework Equations Dot product of vectors The...
  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

    Software for multiplication of matrices I'm going to do a lot of matrix multiplications, since I'm computing Jarlskog invariants. I would like to know if there is a great program for doing a lot of matrix multiplications? I tried with Maple but at some point it gives up. My matrices are not...
  14. TheSodesa

    Proving two simple matrix product properties

    Homework Statement Let ##A## be an n × p matrix and ##B## be an p × m matrix with the following column vector representation, B = \begin{bmatrix} b_1 , & b_2, & ... & ,b_m \end{bmatrix} Prove that AB = \begin{bmatrix} Ab_1 , & Ab_2, & ... & , Ab_m \end{bmatrix} If ##A## is represented...
  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

    Hi, I am studying linear algebra from Golub G.H., Van Loan C.F.- Matrix Computations 3rd edition. This book is somewhat old now, but I find it rather comprehensive. I want to study all chapters and answer all problems appear at the end of each section. Here is the first problem from the first...
  19. C

    Matrix multiplication - is this plausible?

    say X = (AB) (B-1 C) B-1 = B inverse (B B-1 = B-1 B = I) then can i write X = AC? just having a brain fart moment. i would appreciate a speedy response, cheers.
  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

    I understand that the cross product, in lay mans terms doesn't exist unless we're in 3 dimensions. When you multiply two matrices together I have been told you get something similar. I hear that this is because a matrix can be treated as a vector. So if we are talking about measurable...
  22. PsychonautQQ

    Understanding Matrix Multiplication Non-Commutativity

    Homework Statement This isn't homework, but I didn't know where else to ask and I have a test in an hour and a half. The study guide says be able to explain why statements like (A+B)^2 = A^2 + 2AB + B^2 are bogus when dealing with matrix's. Is it because (A+B) = (B+A) but (A+B)^2 /= (B+A)^2 or...
  23. ajayguhan

    Matrix Multiplication: Finding A^50 with a Shortcut Method

    Homework Statement A=[1 0 0] [1 0 1] [0 1 0] Find A^50 Homework Equations The Attempt at a Solution I'm sure that we can't multiply it 50 times...it's a tedious process , there must be a short cut
  24. X

    Explain why matrix multiplication is not commutative.

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

    Understanding Matrix Multiplication: A Vector-Based Explanation

    Consider two square matrix A,B each specifying a parallelopiped by three different vector. The x, y, z components are written in column 1, column 2, column 3 respectively. Thus the order of A , B is 3x3. Let C=AB. To get the c11 element of C i do a dot product of row1 of A and column 1 of B...
  26. G

    MHB Problem involving matrix multiplication and dot product in one proof

    The problem is: Let A be a real m x n matrix and let x be in R^n and y be in R^m (n and m dimensional real vector spaces, respectively). Show that the dot product of Ax with y equals the dot product of x with A^Ty (A^T is the transpose of A). The way I went about starting this problem is to...
  27. AJBMuir

    Matrix multiplication, specifically elimination matrices.

    I was wondering if anybody can see where I have gone wrong here? I was given the rule, E2(E1A1)=(E2E1)A1, I can't seem to find my mistake.
  28. S

    Matrix Multiplication and Function Composition

    I am doing linear algebra and want to fully understand it, not just pass the class. I was recently taught matrix multiplication and decided to look up how it works. The good part is that I understand the concept. Matrices are a way of representing linear transformations. So matrix multiplication...
  29. Y

    Matrix multiplication to addition

    I am looking for a transformation that relates a matrix product with a matrix addition, e.g. AB = PA + QB Is there any such transformation? Thnx
  30. Nugso

    Comp Sci Multiplying Matrices in Fortran: How to Create a Function for Any Dimension

    Hi. Write an external function that takes two matrices, multiplies them and returns the resulting matrix. Matrices can be of any dimension. Print the resulting matrix The above is what I have to do. http://paste.servut.us/pgmm This is what I've done so far. But it seems like...
  31. E

    Divide and Conquer, Matrix Multiplication MATLAB

    Problem Statement We are asked to use the following divide and conquer algorithm to get the solution for the multiplication of some matrix A and some matrix B. (See below) Consider the matrix sizes. Comment the total computational time used on the following three algorithms, when different...
  32. D

    Matrix multiplication: index / suffix notation issues

    Hey everyone, I'm struggling with the summation notation for matrices and vector operations, multiplication in particular. Please refer to the image below where I've typed it all out in Word, its too cumbersome here and I want my meaning to be clear...
  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

    I'm having problems running cblas_dgemm on a matix matrix multiplication. I have the following matricies double * mass = new double[n]; double (* pos)[NDIM] = new double[n][NDIM]; double tempPos[NDIM]; double tempMass[nlocal]; double mass_avg[1]; double pos_avg[NDIM]...
  36. Y

    MHB Matrix Multiplication: Is A Solution Possible?

    Hello I have a question, I need to tell if there exist A such as: A\cdot \begin{pmatrix} 0\\ 1\\ 4 \end{pmatrix} =\begin{pmatrix} 1\\ 2\\ 3\\ 4 \end{pmatrix} how do you approach this kind of questions ? thanks !
  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

    Hello, I have read several different sources on this very topic, and the one thing that confused a little was defining it using sigma notation. Could some please explain to be what it means?
  39. A

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

    1. Why is the norm of a vector noted by double pipes when it is just the magnitude which is notated by single pipes? 2. Does anyone know where I could find out why matrix multiplication is defined the way it is? I know how to do it, but I do not understand why it is that way. Thank you.
  40. R

    Proof Involving Matrix Polynomials and Matrix Multiplication

    Homework Statement Let A be an nxn matrix, and C be an mxm matrix, and suppose AB = BC. (a) Prove the following by induction: For every n∈ℕ, (A^n)B = B(C^n). What property of matrix multiplication do you need to prove this? Homework Equations The four basic properties of matrix...
  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

    !Euler transform matrix multiplication help! Homework Statement This may be rather simple but i am really struggling to complete a 3 3x3 matrix multiplication. I NEED STEP BY STEP WORKING!. This would really help me I understand the theory. Basically I have three matrices T1= cosψ sin ψ...
  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

    Homework Statement true or false .. The system {S,+,.} with S = { matrix (a,b,a-b,a)|a,b ∊ R) is not a field under matrix addition (+) and matrix multiplication (.) i find that the statement is false . since : 1. {S,+} is Abelian group. 2. {S,.} : is Abelian group. is...
  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

    Dear Forum, I have one question on matrix multiplication. Suppose there are 2 matrices - A = 1 -1 0 0 2 -1 2 0 -1 B = 1 1 2 and AB = 0 (Zero Matrix) if B not a zero-matrix, then rank(A) is less than s, where s is the dimension of B. I wanted to...