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

In summary, the highlighted section discusses the meaning of "adjacent indices" in the context of matrix multiplication. The speaker clarifies that the observation is not specific to matrices and applies to any summation, where the summation index is eliminated in the result. They provide an example to illustrate this concept and mention the possibility of the word "adjacent" being used due to the summation convention.
  • #1
Oppie
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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,

upload_2016-10-16_15-5-22.png


Can someone elaborate on the highlighted section? In particular, what does "adjacent indices" mean?

Thank you.
 
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  • #2
'Elimination' just means that the result of the summation does not contain the index over which summation occurs. We see that from the fact that the RHS of the equation, which is just ##y_i##, has no ##k## in it.

The word 'adjacent' is unnecessary in the sentence. Nor is the observation relevant only to matrices. For any summation, the summation index is annihilated in the result. Consider for instance:
$$\sum_{k=0}^nx^k=\frac{1-x^{n+1}}{1-x}$$
There is no ##k## on the right-hand side.
 
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  • #3
andrewkirk said:
'Elimination' just means that the result of the summation does not contain the index over which summation occurs. We see that from the fact that the RHS of the equation, which is just ##y_i##, has no ##k## in it.

The word 'adjacent' is unnecessary in the sentence. Nor is the observation relevant only to matrices. For any summation, the summation index is annihilated in the result. Consider for instance:
$$\sum_{k=0}^nx^k=\frac{1-x^{n+1}}{1-x}$$
There is no ##k## on the right-hand side.

Thank you. Maybe he mentions the word "adjacent" in consideration to the summation convention.
 

What is matrix multiplication?

Matrix multiplication is a mathematical operation that takes two matrices as input and produces a new matrix as output. It involves multiplying each element of one matrix by the corresponding element in the other matrix and then adding the products together to get the resulting element in the new matrix.

Why is matrix multiplication important?

Matrix multiplication is an essential operation in many fields of science and technology, including physics, engineering, computer science, and economics. It allows us to represent and manipulate complex systems and data, making it easier to solve problems and make predictions.

What are the properties of matrix multiplication?

Matrix multiplication has several properties, including associativity, distributivity, and non-commutativity. These properties allow us to simplify and manipulate equations involving matrix multiplication, making it a powerful tool in mathematical and scientific calculations.

How do you perform matrix multiplication?

To perform matrix multiplication, we first need to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. Then, we multiply each element of the first row of the first matrix by each element in the first column of the second matrix, and add the products together to get the first element in the resulting matrix. We repeat this process for each element in the resulting matrix until all elements have been calculated.

What are some common applications of matrix multiplication?

Matrix multiplication is used in a variety of applications, including image processing, data analysis, and machine learning. It is also used in physics to represent and solve systems of equations, and in economics to model and analyze complex economic systems. Additionally, matrix multiplication is used in computer graphics to transform and manipulate images and 3D objects.

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