Orthogonal & Unitary Matrices

Orthonormal columns, Q^TQ = I, and preserving every length.

The idea

An orthogonal matrix is a real square matrix whose columns form an orthonormal set: each column a unit vector, any two perpendicular. In the plane the two columns $q_{1}$ and $q_{2}$ are perpendicular unit vectors, so their tips lie on the unit circle. These are the matrices that move space rigidly, as a rotation or a reflection does, preserving every length and every angle.

The condition on the columns has a compact matrix form. The $(i, j)$ entry of $Q^{T}Q$ is the dot product of column $i$ with column $j$, and orthonormality asks those dot products to be $1$ on the diagonal and $0$ off it. So a real square matrix $Q$ is orthogonal exactly when $Q^{T}Q = I.$ The geometry follows from this equation: $\|Qx\|^{2} = x^{T}Q^{T}Qx = x^{T}x = \|x\|^{2}$, so $Q$ preserves the length of every vector.

Over the complex numbers the transpose becomes the conjugate transpose: $U$ is unitary when $U^{*}U = I$.

Ways to work on it

• Walkthrough. The condition Q^TQ = I, its geometry, and the consequences Q^-1 = Q^T and Q = ± 1.
• Practice. Decide whether a given matrix is orthogonal.
• Hardest. Complete a column to an orthogonal matrix and use length preservation.

Not sure where to start? Take the ten-question placement test.