Orthogonal and Unitary Matrices
Rigid transformations, free inverses, and the gateway to the spectral theorem
1. Definition — Three Equivalent Faces
Recall that the Gram–Schmidt process takes any basis and produces an orthonormal set — vectors that are mutually perpendicular and each of unit length. If you package those orthonormal vectors as the columns of a square matrix, you get an orthogonal matrix.
This single geometric requirement has two powerful algebraic equivalents. The three faces are:
Face 1 — Columns
The columns of $A$ form an orthonormal family.
Face 2 — Product
$A^{\!\top}A = I$ (and also $AA^{\!\top} = I$).
Face 3 — Inverse
$A^{-1} = A^{\!\top}$.
Why the three are equivalent
Face 1 ⟹ Face 2. Write $A = [\mathbf{a}_1 \;|\; \mathbf{a}_2 \;|\; \cdots \;|\; \mathbf{a}_n]$. The $(i,j)$-entry of $A^{\!\top}A$ is $\mathbf{a}_i \cdot \mathbf{a}_j$. Orthonormality says this equals $1$ when $i=j$ and $0$ otherwise — exactly the identity matrix.
Face 2 ⟹ Face 3. If $A^{\!\top}A = I$, then by definition $A^{\!\top}$ is a left inverse of $A$. For square matrices a left inverse is the inverse, so $A^{-1} = A^{\!\top}$.
Face 3 ⟹ Face 1. If $A^{-1} = A^{\!\top}$, then $A^{\!\top}A = I$, and reading off entries gives $\mathbf{a}_i \cdot \mathbf{a}_j = \delta_{ij}$, i.e., the columns are orthonormal.