Linear Algebra - Matrix
m x n matrix
\[A = [a_{ij}] = \begin{bmatrix} a_{11} & a_{12} & . & a_{1n} \\ a_{21} & . & . & . \\ . & . & . & . \\ a_{m1} & . & . & a_{mn} \\ \end{bmatrix}\]matrix multiplication
X = aA + bB + cC + dD, 结果的 nrow = 前矩阵nrow, ncol = 后矩阵ncol.
identity matrix:
\[I = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\]dot/scalar product
\[A = [a_1, a_2, ..., a_n], \ \ \ \ B = \begin{pmatrix} b_1 \\ b_2 \\ . \\ b_n \\ \end{pmatrix} \\ A \cdot B = a_1 b_1 + a_2 b_2 + ... + a_n b_n\]multiply examples
\[c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \\ \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & -1 \\ \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 2 & -2 \\ 1 & 1 \\ \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 7 & -9 \\ \end{bmatrix}\]\[\begin{bmatrix} 1 & -1 & 2 \\ 0 & -3 & 1 \end{bmatrix} \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}\]matrix-vector multiplication
linear equotion
\[2x - 3y = 5 \\ x + 4y = -7 \\ \Rightarrow A \vec{x} = \vec{b} \\ A = \begin{bmatrix} 2 & -3 \\ 1 & 4 \\ \end{bmatrix} , \vec{x} = \begin{pmatrix} x \\ y \\ \end{pmatrix} , \vec{b} = \begin{pmatrix} 5 \\ -7 \\ \end{pmatrix}\]invertible matrices
inverse of A : \(A^{-1}\)
\[AB = BA = I_n \\\]If A is not invertible, A is singular matrix.
transpose of a matrix
symmetric : \(A^t = A\)
skew-symmetric : \(A^t = -A\)
conjugate transpose : \(A^*\)