UTS

• Latsol
• Materi Mingguan & Link Internal

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Rumus

• Norm: $||\mathbf{v}||=\sqrt{v_1^2+v_2^2+\cdots +v_r^2}$
• Inner prod:
  • $⟨\mathbf{u},\mathbf{v}⟩=u_1{v}_1+u_2{v}_2+\cdots +u_r{v}_r$
  • $A\cdot B=\mathrm{tr}(A^{T}B)$
• Gram-Schmidt orthogonalization:
  • ${\mathbf{v}}_i={\mathbf{u}}_i-{\sum }_{j=1}^{i-1}\frac{⟨{\mathbf{u}}_i,{\mathbf{v}}_j⟩}{||{\mathbf{v}}_j|{|}^2}{\mathbf{v}}_j$
  • $B_i=A_i-{\sum }_{j=1}^{i-1}\frac{A_i\cdot B_j}{B_j\cdot B_j}{B}_j$
• Orthonormal/normalizing:
  • $\mathbf{u}=\frac{1}{||\mathbf{v}||}\mathbf{v}$
  • $B=||A|{|}^{-1}A$
• Orthogonal: $\mathbf{u}\cdot \mathbf{v}=0$
• QR Decomposition
  • $A=QR$ where $Q$ has orthonormal cols, $R$ upper triangular
  • $Q=[{\mathbf{q}}_1\cdots {\mathbf{q}}_n]$ from normalizing Gram-Schmidt
  • $R:r_{ij}=⟨{\mathbf{a}}_j,{\mathbf{q}}_i⟩,i\le j$
• Basis
  • Col space: Cols of $A$ corresponding to pivot positions of $A$ (from RREF)
  • Row space: Cols of $A^T$ corresponding to pivot positions of $A^T$
• Dimension: Count of basis of vector space
• Rank
  • Dimension of col/row space (equal)
  • Count of pivot cols
  • $\mathrm{rank}(A)=\mathrm{rank}(T)+\mathrm{rank}(W-V{T}^{-1}U)$
• G-Inverse
  1. $r\times r$ invertible submatrix $M$, $r=\mathrm{rank}(A)$
  2. Create $n\times m$ zero matrix, and insert $(M^{-1}{)}^T$ where $M$ was taken from $A$
  3. Matrix from (2) is $(A^{-1}{)}^T$
• Partition
  • Product: Treat partitioned as elements of $A$, and multiply normally, then expand
  • Inverse:
    1. $t=T^{-1}$, $u=tU$, $v=Vt$, $q=(W-vU{)}^{-1}$
    2. ${\left[\begin{array}{cc}T& U\\ V& W\end{array}\right]}^{-1}=\left[\begin{array}{cc}t& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{c}-u\\ I_n\end{array}\right]q\left[\begin{array}{cc}-v& I_n\end{array}\right]$ atau ${\left[\begin{array}{cc}W& V\\ U& T\end{array}\right]}^{-1}=\left[\begin{array}{cc}0& 0\\ 0& t\end{array}\right]+\left[\begin{array}{c}{I}_n\\ -u\end{array}\right]q\left[\begin{array}{cc}{I}_n& -v\end{array}\right]$