Matrix

matrix 는 두 가지 의미로 해석될 수 있다.

  • linear equations

  • linear mapping

Definition

  • $n, m \in R$

  • matrix A 는 $n \cdot m$ tuple of elements: $a_{ij}, i = 1, 2, \cdots, n, j =1, 2, /cdots, m$

  • \begin{equation*} A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix} \end{equation*}
  • column: (1, m)
  • row: (1, n)

Matrix Addition and Multiplication

  • Addition

  • Multiplication(Hadamard product)
  • Multiplication(dot product)
    • Associativity
      • $A(BC) = (AB)C.$
    • Distributivity
      • $A(B + C) = AB + AC$
    • Multiplication with identity matrix
      • $IA = AI = A$

Inverse and Transpose

  • identity matrix

  • Inverse

    • $A, B \in R^{nxn}$
    AB=In=BAAB = I_n = BA

  • Transpose

    • $A \in R^{nxm}, B \in R^{mxn}$
    • $b_{ij} = a_{ij}$
    • $A^T = B$
    • $B^T = A$
  • Symmetric matrix
    • $A^T = A$
  • Some properties
    • $AA^{-1} = I = A^{-1}A$
    • $(AB)^{-1} = B^{-1}A^{-1}$
    • $(A+B)^{-1} \neq A^{-1} + B^{-1}$
    • $(A^T)^T=A$
    • $(A+B)^T = A^T + B^T$
    • $(AB)^T = B^TA^T$
    • $(A^{-1})^T = (A^T)^{-1}$

Multiplication by scalars

  • Associativity

    • $\lambda(\phi C) = (\lambda \phi)C$

  • Distributivity

    • $\lambda(B + C) = \lambda B + \lambda C$

Compact Representations of Systems of Linear Equations

  • linear equation

  • Vector

  • Matrix