if and are perpendicularthen pythagorastheorem holds|| || hx yx yi hxxi hxyi hyyi || || || || pythagorastheorem ( basis { vn of rn in which all the vectors are pairwise perpendicular and have norm is called an orthonormal (short for orthogonal and normalizedbasis for examplethe standard basis is orthonormal orthonormal theorem orthonormal basis representation if { vn is an orthonormal basis of rn then any vector rn can be expressed as hxv hxvn vn ( proofobserve thatbecause the {vi form basisthere exist unique an such that an vn by the linearity of the inner product and the orthonormality of the {vi it follows that hxv ai vi an matrix whose columns form an orthonormal basis is called an orthogonal matrix note that for an orthogonal matrix [ vn ]we have orthogonal matrix > > vn vv [ vn vn vn vn vn vn hencev- vnote also that an orthogonal transformation is length preservingthat isvx has the same length as this follows from length preserving ||vx|| hvxvxi xvvx xx || || complex vectors and matrices instead of the vector space rn of -dimensional real vectorsit is sometimes useful to consider the vector space cn of -dimensional complex vectors in this case the adjoint or conjugate transpose operation (*replaces the transpose operation (>this involves the usual transposition of the matrix or vector with the additional step that any complex number is replaced by its complex conjugate for exampleif xand aa then [ and the qualifier "orthogonalfor such matrices has been fixed by history better term would have been "orthonormaladjoint