Orthogonal Matrix

Description:

• Orthogonality
• Square Matrix
• Columns form Orthonormal Basis
• If $U=[u_1,\dots ,u_n]$ is an orthogonal matrix, then $u_i^{⊺}{u}_j=\{\begin{array}{l}1\text{ if }i=j\\ 0\text{ otherwise.}\end{array}$
  • Thus, $U^{⊺}U=I_n$, Identity Matrix
• As its columns are orthonormal basis, it preserves angles and length of the vectors when multiplied
  • every vector $x$, $||Ux|{|}_2^2=(Ux{)}^T(Ux)=x^T{U}^{T}Ux=x^{T}x=||x|{|}_2^2$
  • rotational invariance of the Euclidean norm
• Produces a rotation
• Tranpose an orthogonal matrix is its inverse