Transpose and Properties

1. The Transpose of a Matrix ($A^T$)

Definition

The transpose of an $(m \times n)$ matrix $A$, denoted by $A^T$, is an $(n \times m)$ matrix obtained by swapping the rows and columns.

• The first row of $A$ becomes the first column of $A^T$.
• The second row of $A$ becomes the second column of $A^T$.

Theorems of the Transpose

$(A^T)^T = A$ (Flipping it twice puts it back to normal)

CRITICAL: $(AB)^T = B^T A^T$ (Notice the order swaps! You cannot just distribute the $T$ without flipping the matrices).

2. Pro-Tips for Exams

The Trace of a Product Shortcut

If an exercise asks you to find the trace of a multiplied matrix, like $\text{tr}(AB)$, do not waste time computing the entire matrix $AB$.

Because the trace is only the sum of the main diagonal, you only need to calculate the main diagonal entries of $AB$ (which are $c_{11}, c_{22}, c_{33}$, etc.) and add them together. Ignore the rest of the matrix!

3. The Identity Matrix and Algebraic Properties

The Identity Matrix ($\mathbb{I}_n$)

A square matrix with 1s on the main diagonal and 0s everywhere else is called an identity matrix. It acts like the number "1" for matrices!

Example ($\mathbb{I}_2$): $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Crucial Property: Any matrix multiplied by the identity matrix remains unchanged. $A \cdot \mathbb{I} = A$ and $\mathbb{I} \cdot A = A$.

Algebraic Properties of Matrix Operations

Assuming matrix sizes allow these operations to be performed, matrix arithmetic follows these rules:

• Commutative (Addition): $A + B = B + A$
• Associative (Addition): $(A + B) + C = A + (B + C)$
• Associative (Multiplication): $(AB)C = A(BC)$
• Distributive (Scalar over Matrix Addition): $c(A \pm B) = cA \pm cB$
• Distributive (Scalar Addition over Matrix): $(a \pm c)A = aA \pm cA$
• Associative (Scalar Multiplication): $(ac)A = a(cA)$
• Scalar Identity: $1 \cdot A = A$