2.9c Matrix Orthonormality

Orthonormal Matrices

If a matrix $Q$…

• is a tall matrix ($Q\in {\mathbb{R}}^{m\times n}$ for $m>n$), then it has orthonormal columns if and only if $Q^{T}Q=I_n.$
• is a wide matrix ($Q\in {\mathbb{R}}^{m\times n}$ for $m<n$), then it has orthonormal rows if and only if $Q{Q}^T=I_m.$
• is a square matrix ($Q\in {\mathbb{R}}^{n\times n}$), then it is orthonormal if and only if $Q^T=Q^{-1}.$

Intuition

Essentially, the above theorem may be translated into set orthonormality:

• A tall matrix has orthonormal columns $⟺$ its column vectors form an orthonormal set.
• A wide matrix has orthonormal rows $⟺$ its row vectors form an orthonormal set.
• A square matrix is orthonormal $⟺$ both its row and column vectors form orthonormal sets.

As such, the product of the matrix and its transpose is equal to the identity matrix $I$ since the pairwise dot product of its columns/rows is 0 when two different columns/rows are multiplied and 1 when the same column/row is multiplied. This leads to an identity matrix.

The last equality simply shows that both of the above hold, since $Q{Q}^{-1}=Q^{-1}Q=I_n$.

Properties

(Preservation of Length and Angle) $\Vert Qx{\Vert }_2=\Vert x{\Vert }_2$.

(Projection Onto Range) If $Q^{T}Q=I_n$, then $Q{Q}^T$ is a projection onto $\text{Col}(A)$.