03 Matrix vs Element-by-Element Arithmetic

Overview: Algebraic Math vs. Vectorized Math

MATLAB features two distinct mathematical systems: Matrix Arithmetic (which treats arrays as whole linear systems) and Element-by-Element Arithmetic (which applies operations element-wise).

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1. Matrix Operators vs. Element-wise Operators

Operators preceded by a period (.) run element-wise, whereas standard operators perform matrix linear algebra operations.

Operator	Type	Math Description	Dimension Requirements
A * B	Matrix	Traditional dot product matrix multiplication.	Inner dimensions must match: $(M\times N)\times (N\times P)$
A .* B	Element-wise	Multiplies individual elements: $C(i,j)=A(i,j)\cdot B(i,j)$.	Dimensions of $A$ and $B$ must match exactly (or broadcast)
A / B	Matrix	Right division: equivalent to $A\cdot B^{-1}$.	Column count of $A$ must equal column count of $B$
A ./ B	Element-wise	Divides individual elements: $C(i,j)=A(i,j)/B(i,j)$.	Dimensions of $A$ and $B$ must match exactly
A ^ n	Matrix	Matrix power: multiplies $A$ by itself $n$ times ($A\cdot A\dots A$).	$A$ must be a square matrix ($N\times N$)
A .^ n	Element-wise	Raises each individual element to power $n$.	Any dimension shape allowed

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2. Element-wise Functions vs. Matrix Functions (m suffix)

MATLAB has parallel function suites for scalar and matrix math. Functions ending with m operate on the matrix as a whole system.

Dynamic Comparisons:

Exponentiation:

• exp(A)
  • Syntax: Y = exp(A)
  • Behavior: Computes $e^{A(i,j)}$ for every element.
• expm(A)
  • Syntax: Y = expm(A)
  • Behavior: Computes the matrix exponential via Taylor series expansions: $e^A=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\dots$

Square Root:

• sqrt(A)
  • Syntax: Y = sqrt(A)
  • Behavior: Computes the square root of each individual element.
• sqrtm(A)
  • Syntax: X = sqrtm(A)
  • Behavior: Computes the matrix square root, finding a matrix $X$ such that $X^2=X\cdot X=A$.

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💻 Conceptual Code Demonstration

% 1. Create a Matrix
A = [1 2; 3 4];
B = [2 0; 1 2];
 
% 2. Compare Multiplications
matrixMul = A * B;   % Result: [4 4; 10 8] (Matrix multiplication)
elementMul = A .* B; % Result: [2 0; 3 8] (Element-by-element multiplication)
 
% 3. Compare Powers
matrixPower = A ^ 2;   % Result: [7 10; 15 22] (A * A)
elementPower = A .^ 2; % Result: [1 4; 9 16] (Element-by-element squaring)
 
% 4. Matrix Exponential vs Element Exponential
elementExp = exp(A);  % Calculates exp(1), exp(2), etc.
matrixExp = expm(A);  % Calculates e^A using matrix algebra