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    Orthogonality of a matrix via its columns

    E-HK-104-4

    Area
    Algebra
    Topics
    Groups Preserving Bilinear Forms, Linear Algebra, Orthogonality
    Status
    draft

    Let \(M\) be an \(n \times n\) matrix over \(C\) with columns \(M_1, M_2, \ldots, M_n\). Show that \(M\) belongs to \(O(n, C)\) if and only if \[ M_j^tM_k = \delta_{jk}. \]

    Solution

    Let \(M_j\) denote the \(j\)th column of \(M\). The \((j,k)\) entry of \(M^tM\) is \[ (M^tM)_{jk} =\sum_{r=1}^n M_{rj}M_{rk} =M_j^tM_k. \] Therefore \[ M^tM=I \] if and only if \[ M_j^tM_k=\delta_{jk} \] for every pair \(j,k\).

    But \(M\in O(n,\mathbb C)\) is defined exactly by \(M^tM=I\). Hence \[ \boxed{M\in O(n,\mathbb C)\iff M_j^tM_k=\delta_{jk}.} \]

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