Can you explain the difference between the two matrix operations in MATLAB?

I am trying to understand how MATLAB performs the following matrix operations:

Example 1:

clearvars; clc; close all;
Nx = 8; 
Ny = 8;
Lx=2*pi;
dx = Lx/Nx;
Vec  = fftshift(-Nx/2:Nx/2-1);
Vector1 = (sin( Vec * dx/2)/(dx/2)).^2 ;
 
[Matrix2,x] = cheb(Ny);
 for m = 1:length(Vec)
        Matrix1 = -1 * (Vector1(m))+ Matrix2;
 end
 

Example 2:

clearvars; clc; close all;
Nx = 8; 
Ny = 8;
Lx=2*pi;
dx = Lx/Nx;
Vec  = fftshift(-Nx/2:Nx/2-1);
Vector1 = (sin( Vec * dx/2)/(dx/2)).^2 ;
Igl = speye(Ny+1);
[Matrix2,x] = cheb(Ny);
 for m = 1:length(Vec)
        Matrix1 = -Igl * (Vector1(m))+ Matrix2;
    end

Why is Matrix1 different in Example1 and Example 2? In particular, in Example 1 how is the scalar multiplication of the row vector (Vector1(m)) added to Matrix 2? I am trying to understand the matrix operation done in Example 1 specifically so I can transfer it to C/C++. Thanks

Answer 1

[Saurav]

Hi Janee,

I see you want to understand how each example handles the multiplication and addition operations in a matrix.

Example 1:

• The loop iterates through each element of Vec.
• For each iteration, Matrix1 is set to -1 * (Vector1(m)) + Matrix2.
• This means that during each iteration, Matrix1 is overwritten with the result of the scalar -1 * (Vector1(m)) added to the matrix Matrix2.
• Because Matrix2 is a matrix and -1 * (Vector1(m)) is a scalar, MATLAB will broadcast the scalar across all elements of Matrix2. This effectively adds the scalar to each element of Matrix2.

Example 2:

• The loop is similar, but this time -Igl * (Vector1(m)) is used instead of -1 * (Vector1(m)).
• Here, Igl is the identity matrix of size (Ny+1) x (Ny+1).
• When you multiply the scalar Vector1(m) by Igl, you get a diagonal matrix where each diagonal element is -Vector1(m).
• This diagonal matrix is then added to Matrix2.

Key Differences:

• Scalar Addition vs. Matrix Addition: In Example 1, a scalar is added to every element of Matrix2. In Example 2, a diagonal matrix (with -Vector1(m) on the diagonal) is added to Matrix2.
• Resulting Matrix: In Example 1, Matrix1 is a matrix where each element is Matrix2(i,j) - Vector1(m). In Example 2, Matrix1 is a matrix where the diagonal elements are Matrix2(i,i) - Vector1(m) and the off-diagonal elements are the same as in Matrix2.

Refer to the following documentation to learn more about arrays and matrix operations:

https://www.mathworks.com/help/matlab/matlab_prog/array-vs-matrix-operations.html

Hope this helps!