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Module 2: Quick Reference Card

Matrix Algebra -- Linear Algebra

Matrix Operations

Addition / Scalar

  • Same size required for +
  • (A+B)_{ij} = a_{ij} + b_{ij}
  • (cA)_{ij} = c * a_{ij}

Multiplication

  • (m x n)(n x p) = m x p
  • (AB)_{ij} = row i of A dot col j of B
  • NOT commutative

Key Warnings

AB != BA    |    AB=0 does not mean A=0 or B=0    |    AB=AC does not mean B=C

Transpose

Definition

(A^T)_{ij} = a_{ji}. Rows become columns.

Properties

  • (A^T)^T = A
  • (AB)^T = B^T A^T
  • A^T = A means symmetric

Inverse

2x2: [a b; c d]^{-1} = (1/(ad-bc)) [d -b; -c a]

General Method

Row reduce [A|I] to [I|A^{-1}]

Properties

  • (AB)^{-1} = B^{-1}A^{-1}
  • x = A^{-1}b solves Ax=b

LU Factorization

A = LU  |  L = lower tri (multipliers)  |  U = upper tri (REF)

Solve Ax=b: (1) Ly=b forward sub, (2) Ux=y back sub.