Orthogonal and Orthonormal Matrices

Definition

A real square matrix $Q$ is orthogonal (the convention varies; some authors say "orthonormal") if its columns form an orthonormal set. Equivalently: $Q^T Q = Q Q^T = I$

which means $Q^{-1} = Q^T$ — the inverse is just the transpose.

What the condition means, column by column

If $Q$'s columns are $\mathbf{q}_1, \mathbf{q}_2, \ldots, \mathbf{q}_n$, then $(Q^T Q)_{ij} = \mathbf{q}_i \cdot \mathbf{q}_j$. So $Q^T Q = I$ is equivalent to

• $\mathbf{q}_i \cdot \mathbf{q}_j = 0$ for $i \ne j$ (orthogonal columns), and
• $\mathbf{q}_i \cdot \mathbf{q}_i = 1$ (unit length) — the columns are orthonormal.

Step-by-step — rotation matrix

$Q = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$

Step 1 — Compute $Q^T Q$. $Q^T = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$

$Q^T Q = \begin{pmatrix} \cos^2\theta + \sin^2\theta & -\cos\theta \sin\theta + \sin\theta \cos\theta \\ -\sin\theta \cos\theta + \cos\theta \sin\theta & \sin^2\theta + \cos^2\theta \end{pmatrix}$

$= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I ✓$

Step 2 — Check $Q Q^T$ (should also equal $I$ for square $Q$).

By similar calculation, $Q Q^T = I$ ✓.

$Q$ is orthogonal.

Length preservation

For any $\mathbf{x}$: $\|Q \mathbf{x}\|^2 = (Q \mathbf{x})^T (Q \mathbf{x}) = \mathbf{x}^T Q^T Q \mathbf{x} = \mathbf{x}^T \mathbf{x} = \|\mathbf{x}\|^2$

So $\|Q \mathbf{x}\| = \|\mathbf{x}\|$ — orthogonal matrices are isometries.

Angle preservation

$(Q \mathbf{u}) \cdot (Q \mathbf{v}) = \mathbf{u}^T Q^T Q \mathbf{v} = \mathbf{u}^T \mathbf{v}$

The dot product is preserved — hence angles between vectors are preserved.

Together, length and angle preservation mean orthogonal matrices are rigid motions fixing the origin: rotations and reflections.

Orthogonal = rotation or reflection

A real orthogonal matrix always has $\det Q = \pm 1$:

• $\det Q = +1$: $Q$ is a rotation (preserves orientation). These form $SO(n)$.
• $\det Q = -1$: $Q$ is a reflection (reverses orientation). Together with rotations they form $O(n)$.

Key properties

• Inverse is transpose: $Q^{-1} = Q^T$. Cheap to compute.
• Product of orthogonal is orthogonal: if $P, Q$ are orthogonal, so is $PQ$.
• Eigenvalues lie on the unit circle (modulus 1): $|\lambda| = 1$ for every eigenvalue.
• Determinant: $|\det Q| = 1$.

Why orthogonal matrices are amazing

• Numerically stable. In floating-point, errors don't amplify under orthogonal multiplication.
• Easy inverse. Transposing is $O(n^2)$ vs. general inversion at $O(n^3)$.
• Building block of decompositions. QR, SVD, and spectral theorem all use orthogonal/unitary matrices.
• Physics: represent symmetries (rotations of coordinate frames, spin, etc.).
• Computer graphics: rotations preserve shape; reflections mirror it.

Orthogonal vs. orthonormal matrix?

Terminology varies. Strictly:

• Orthonormal matrix: columns are orthonormal (orthogonal and unit length). Same as our definition above.
• "Orthogonal matrix": most linear algebra texts use this for the orthonormal case.
• Some applied contexts use "orthogonal" loosely to mean just orthogonal (not unit-length) columns, but that's non-standard.

The rule of thumb: "orthogonal matrix" ⟹ $Q^T Q = I$.

Common mistakes

• Checking only orthogonality, not unit length. A matrix with perpendicular but non-unit columns is not orthogonal.
• Forgetting $\det Q = \pm 1$. If the determinant is not one of these, the matrix is not orthogonal.
• Assuming all symmetric matrices are orthogonal. They're different: symmetric means $A^T = A$; orthogonal means $A^T A = I$.

Try it in the visualization

Rotate $\theta$ through $0$ to $2\pi$. Columns of $Q$ traverse the unit circle; $Q^T Q$ stays as identity (all four entries display $1, 0, 0, 1$). A unit square rotates without distortion — confirming length and angle preservation.