3. Array operations and Linear equations

MATLAB has two different types of arithmetic operations: matrix arithmetic operations and array arithmetic operations. We have seen matrix arithmetic operations in the previous lab. Now, we are interested in array operations.

In MATLAB, there are two types of arithmetic operations that can be performed on data structures: matrix arithmetic and array arithmetic. Matrix arithmetic involves operations on matrices themselves, while array arithmetic operates on the individual elements of the matrices. Both types of operations are crucial in manipulating data elements effectively.

Think of matrix operations like cooking a dish, where you mix several ingredients together according to a recipe, producing a single final dish (matrix). In contrast, array operations are like preparing individual ingredients separately, allowing you to adjust each ingredient (element) individually before mixing them, leading to a personalized final dish.

MATLAB allows arithmetic operations: +, −, , and ˆ to be carried out on matrices. Thus, A+B or B+A is valid if A and B are of the same size AB is valid if A’s number of column equals B’s number of rows A^2 is valid if A is square and equals AA αA or A*α multiplies each element of A by α.

Matrix arithmetic allows specific operations based on the dimensions of the matrices involved. Addition and subtraction can be performed if the matrices are of the same size, while multiplication is contingent on the number of columns in the first matrix matching the number of rows in the second. Raising a matrix to a power requires the matrix to be square. Additionally, multiplying a matrix by a scalar means multiplying every element of the matrix by that scalar.

Imagine you are organizing two sets of books (matrices). You can only stack two sets on top of each other if they have the same number of books (size). If you plan to categorize them, you can only do it if one set is arranged in columns and the other in rows that match up logically (matrix multiplication). If you want to label each book with a rating (scalar multiplication), each book receives the same label.

Array arithmetic operations or array operations for short, are done element-by-element. The period character, ., distinguishes the array operations from the matrix operations. However, since the matrix and array operations are the same for addition (+) and subtraction (−), the character pairs (.+) and (.-) are not used. The list of array operators is shown below in Table 3.1.

Array operations involve performing arithmetic on individual elements rather than operating on the matrix as a whole. This is indicated by using a period (.) before the operation symbol. For example, using .*, ./, and .^ means that each corresponding element from two matrices is multiplied, divided, or exponentiated separately. This allows for more granular control when manipulating matrix data.

Think of array arithmetic operations as a factory where each worker is responsible for assembling individual parts of a product (elements of the matrices). Each worker (element in the array) can perform their tasks without waiting for the others—making it efficient to produce complex products without impacting the entire operation.

If A and B are two matrices of the same size with elements A = [a_ij] and B = [b_ij], then the command C = A.*B produces another matrix C of the same size with elements c_ij = a_ij * b_ij.

This example illustrates how element-by-element multiplication works. Given two matrices A and B of equal size, when we perform the operation 'C = A .* B', we create a new matrix C where each element is the product of the corresponding elements from A and B. For instance, if matrix A has an element at position (1, 1) that is 1 and matrix B has (1, 1) as 10, then C(1, 1) will be 1 * 10 = 10.

Imagine you are planning a party and you have a list of guests (matrix A) and the number of dishes each one is supposed to bring (matrix B). By multiplying the two lists together, you would come up with the total number of dishes you are expecting (new matrix C), ensuring each guest's contribution is accurately accounted for in the final tally.

One of the problems encountered most frequently in scientific computation is the solution of systems of simultaneous linear equations. With matrix notation, a system of simultaneous linear equations is written Ax = b.

In scientific computation, solving systems of linear equations is a common task. A system can be represented in matrix form as 'Ax = b', where A is the matrix of coefficients, x is the vector of variables we need to solve for, and b is the result vector. The goal is to find the vector x that satisfies this equation.

Think of a group project where several students (the variables in vector x) need to complete tasks together (the equations). The distribution of tasks among them can be represented in a chart (matrix A), and their collective goal (the target outcomes) is outlined in another list (vector b). The task is to determine how to allocate responsibilities effectively (solve for x).

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