Resolution of the identity in an orthonormal basis ($I={\sum }_{j=1}^n{u}_j{u}_j^T$)

For a real orthonormal basis $u_1,\dots ,u_n$, each vector has expansion $x={\sum }_j{u}_j(u_j^{T}x)$. Since this holds for every $x$, the sum of the rank-one orthogonal projection matrices $u_j{u}_j^T$ is the identity matrix. Their pairwise products vanish for distinct basis vectors. For a complex inner product space, replace the transpose by the conjugate transpose.