MATH 110 - Day 11

2024-10-02 10:05:38 -0400 EDT

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Transposing Matrixes

Let A be an $ n \times m$ matrix

Definition: The Transpose is denoted by $A^T$, is the matrix $ n \times m$ becoming $ m \times n$ (swaps rows and columns)

i.e. $ B = \left[ \begin{array}{r} 2 \\ 4 \end{array} \right], \ B^T = [2 \ 4 ] $

If $A$ is a square matrix such that $A = A^T$, then $A$ is symmetric (all the numbers match along the diagonal).

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Algebra for Matrix Operations

Addition between Two Matrices:

1. Commutative: $A + B = B + A$
2. Associative: $A + (B + C) = (A + B) + C$
3. Distributive: $ c(A + B) = cA + cB$

Matrix Multiplication:

1. Associative: $A(BC) = (AB)C$
2. Distributive: $A(B + C) = AB + AC$
3. Also: $k(AB) = (kA)B = A(kB)$
4. Also: $ I_nA = AI_n$

Transposing a Matrix:

1. $(A^T)^T = A$
2. $(kA)^T = kA^T$
3. $(A + B)^T = A^T + B^T$
4. $(AB)^T = B^TA^T$ (it gets flipped for some reason!)

Note: $(AA^T)^T = AA^T \Rightarrow AA^T$ is symmetric

Note: $(A + A^T)^T = A + A^T \Rightarrow A + A^T$ is symmetric