3.1.2. Array arithmetic operations

On the other hand, 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.

Array arithmetic operations in MATLAB are fundamentally different from matrix operations. Instead of applying the operation to the whole matrix at once, array operations process each corresponding element individually. This feature is denoted in MATLAB with a period before the operator, indicating that the operation applies to each element in the arrays. Notably, for addition and subtraction, the usual operators (+ and −) can still be used without the period.

Imagine you have two baskets of fruits with the same number of items. When you perform array operations, you treat each item in the basket separately, say adding apples to apples, oranges to oranges. Just like when counting apples and oranges separately, you can directly apply arithmetic to each fruit in the respective baskets.

The list of array operators is shown below in Table 3.2. 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.

In MATLAB, array operators allow you to perform specific operations on each corresponding element of two arrays or matrices. For example, if you have two 3x3 matrices A and B, using the command C = A.*B will create a new matrix C where each element c_ij is the product of the elements a_ij from matrix A and b_ij from matrix B. This signifies multiplication tailored specifically to each pair of elements.

Consider cooking where you have two recipes, one for cookies and one for cupcakes, with the same number of ingredients. If you decide to double each ingredient for both recipes, array operators would let you compute the total needed ingredient for both simultaneously, resulting in two new batches of dough—one for cookies and one for cupcakes—without mixing them up.

For example, using the same 3 x 3 matrices, A = [1 2 3; 4 5 6; 7 8 9], B = [10 20 30; 40 50 60; 70 80 90], we have, >> C = A.*B C = [10 40 90; 160 250 360; 490 640 810]

This example demonstrates how array operations work in practice. When multiplying matrices A and B element by element, MATLAB returns a new matrix C where each element is derived from A and B's corresponding elements. For instance, the first element of C (c_11) equals the first element of A (1) multiplied by the first element of B (10), yielding 10.

Think of this like a game of matching colors where you combine matching cups of paint from two tables. If table A has cups of red paint and table B has cups of blue paint, for every red cup (1) matched with a blue cup (10), you create a new mixed cup (10). Continuing this process for all cups produces a new set of colors based on your combinations.

To raise a scalar to a power, we use, for example, the command 10^2. If we want the operation to be applied to each element of a matrix, we use .^2. For example, if we want to produce a new matrix whose elements are the square of the elements of the matrix A, we enter >> A.^2 ans = [1 4 9; 16 25 36; 49 64 81]

When you want to apply an exponentiation operation across each element of a matrix, you utilize the array exponentiation operator (.^). This allows each individual element in A to be squared, resulting in a new matrix where each position contains the square of the original element. For example, squaring each number in the matrix A transforms 1 to 1, 2 to 4, and so on.

Imagine you are planting seeds in a garden, and every type of flower you plant grows a certain number of flowers based on how many you initially planted. If you planted 1 seed and it produces 1 flower, 2 seeds produce 4 flowers (2^2), and so forth. Each item grows independently but follows the same growth pattern.

The relations below summarize the above operations. To simplify, let’s consider two vectors U and V with elements U = [u_i] and V = [v_i]. U .* V produces [u_1v_1, u_2v_2, ..., u_n*v_n]. U ./ V produces [u_1/v_1, u_2/v_2, ..., u_n/v_n]. U .^ V produces [u_1^v_1, u_2^v_2, ..., u_n^v_n].

This section emphasizes the pattern of array operations' outcomes based on the operations performed. Arrays can be combined in a variety of ways, such as through multiplication, division, and exponentiation, with each operation producing a new array where each element relates directly to the corresponding elements of the original arrays. This allows for flexibility and precision in calculations.

Think of it as a team of workers where each member has a set of tasks (elements in U and V). When each performs their tasks together (array operations), like multiplying or dividing, they complete a project efficiently, producing outcomes based on their combined work (new array) rather than just their individual performances (matrix operations).

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