Home/Algebra/Orthogonality and projections/Orthogonal matrix21. Orthogonality and projectionsOrthogonal matrixLevel: intermedioIn 21. Orthogonality and projectionsFormulaCopy LaTeXQTQ=I⇒Q−1=QTQ^TQ=I\Rightarrow Q^{-1}=Q^TQTQ=I⇒Q−1=QTTagsalgebraalgebra formulalinearorthogonalityorthogonality projectionsDetailPreserves lengths and angles.Explore the ideaFound an error? Contact us.Related formulasRelatedTranspose(AT)ij=aji;(AB)T=BTAT(A^T)_{ij}=a_{ji};\quad(AB)^T=B^TA^T(AT)ij=aji;(AB)T=BTATOpen formulaOpen in sectionRelatedSpectral theoremA=AT⇒A=QΛQTA=A^T\Rightarrow A=Q\Lambda Q^TA=AT⇒A=QΛQTOpen formulaOpen in sectionRelatedQR decompositionA=QRA=QRA=QROpen formulaOpen in section