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Matrix

#Linear Algebra

Properties

  • \(A (BC) = (AB) C\): Associative Law (結合)
  • \(AB \neq BA\): Communitative Law (交換)
  • \(A (B+C) = Ab + AC\)；\((A+B) C = AB + AC\): Distributitive Law (分配)

Matrix: with Gaussian-Jordan Elimination

  • use vector-matrix product to represent row reduction (or adding) in Elimination

    matrix-vector product:

    • 如果是 matrix-column-vector product，結果就是 the linear combination of columns of the matrix
      • \(A_{m\times n}\boldsymbol{v_{n\times 1}} = \boldsymbol{b_{m\times 1}}\) (column vector 在右)
    • 若為 matrix-row_vector product，結果是 the linear combination of rows of the matrix
      • \(\boldsymbol{v_{1\times m}A_{m\times n}} = \boldsymbol{b_{1\times n}}\) (row vector 在左)
    • vector entries are the coefficients

    上述可以推廣至 matrix multiplication

Elucidation

Matrix 的每一 row 皆為一 linear equation，利用 matrix-row vector product 來表示某一種列運算 (the linear combination of rows)；因為列運算只包含加、乘(scalar)，符合 linear combination 的定義。

Summary

Gaussian Elimination of any matrix can be expressed as multiplications of matrices using only elimination and permutation matrices.

  • Elimination matrix: \(E_{ij}\) for row deduction

    • Reduce row \(i\) with row \(j\)

  • Permutation matrix: \(P_{ij}\) for row exchange

    • Exchange row \(i\) and row \(j\) (in identity matrix \(I\))

      \(P_{ij} \times P_{ij} = I\)
      e.g. \(P_{21} = \begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}\)

Inverse

Definition

\(A\) is invertible if there exists a matrix \(B\) s.t. \(AB = BA = I\), and \(B\) is the inverse matrix of \(A\) (\(A^{-1}\)).

Note: Both right(\(AB=I\))and left(\(BA=I\))inverse must exist.

Theorem

Invertible matrix must be a square matrix. (Both left inverse and right inverse exists)

Gaussain-Jordan Elimination for Finding Inverse Matrix

If \([\ A\ | \ B\ ] \rightarrow [\ I\ | \ R\ ]\) is obtainable, then \(A^{-1} = R\).

Transpose

For matrix \(A\), denote \(A^T\) as the transpose of \(A\), where the columns of \(A^T\) are the rows of \(A\).

Theorem

  1. \[(A+B)^T = A^T + B^T\]
  2. \[(AB)^T = B^TA^T\]
  3. \[(A^{-1})^T = (A^T)^{-1}\]

Symmetric

If \(A\) is symmetric, \(A = A^T\), i.e. \(a_{ij} = a_{ji}\).

性質頗多，留待下次。