Orthogonality & Orthogonal Matrices

The last lesson promised a matrix whose action keeps lengths and angles intact — rigid, like turning a sheet of paper on a desk. The shapes drawn on it shift, but nothing stretches and no right angle bends. That is an orthogonal matrix: it moves vectors about without ever changing a length or an angle. Its whole family is the rigid motions — rotations and reflections, nothing fancier — and the tidy Aᵀ property the last lesson hinted at falls straight out.

Start with the word alone: two vectors are orthogonal when their dot product is zero, u · v = 0 — they meet at a right angle. An orthogonal matrix is the natural upgrade: a whole basis of directions that are mutually perpendicular and unit length.

Orthonormal columns, and QᵀQ = I

A square matrix Q is orthogonal when its columns are orthonormal — each a unit vector, any two of them perpendicular. Stack that condition into matrix form and it collapses to one clean equation:

Because QᵀQ = I, the inverse is simply the transpose: Q⁻¹ = Qᵀ. Inverting an orthogonal matrix costs nothing — you just flip it, the tidy property promised. (The same holds for the rows, so QQᵀ = I too.)

They keep length and angle

The defining geometric fact is that an orthogonal matrix never changes a length. For every x, the QᵀQ = I in the middle cancels:

‖Qx‖² = (Qx)ᵀ(Qx) = xᵀ QᵀQ x = xᵀ I x = xᵀx = ‖x‖²

so ‖Qx‖ = ‖x‖. The same cancellation keeps dot products intact ((Qx)·(Qy) = x·y), so angles survive as well. A rotation is the clearest picture — the vector spins, its length untouched, both ends tracing the same circle:

In the playground, set the entries to cosθ, −sinθ, sinθ, cosθ (an honest rotation) and the unit shape spins without distorting; drag them off those values and lengths and angles change at once — the visual signature of not being orthogonal.

Determinant and eigenvalues

Take determinants of QᵀQ = I: since det(Qᵀ) = det(Q), we get det(Q)² = 1, so det(Q) = +1 or −1 — +1 a pure rotation, −1 a reflection.

The eigenvalues are subtler. Because lengths are preserved, Qx = λx forces ‖x‖ = ‖Qx‖ = |λ|·‖x‖, so |λ| = 1: every eigenvalue sits on the unit circle in the complex plane. That is not the same as every eigenvalue being ±1 — a genuine rotation has no real eigenvectors at all, and its eigenvalues are a complex conjugate pair cosθ ± i·sinθ. This is the trap GATE sets.

A worked example — the 2025 norm-preserving question

Let A be a real n × n matrix with ‖Ax‖ = ‖x‖ for every x. Which must be true?

Each length preserved means each dot product preserved, hence AᵀA = I, so A is orthogonal (TRUE). That gives an inverse (Aᵀ), so A is invertible / full rank (TRUE). And det(A)² = det(AᵀA) = det(I) = 1, so det(A) = ±1 (TRUE). The trap option, “all eigenvalues are ±1”, is FALSE — the 90° rotation shows why:

A = [ 0  −1 ]      AᵀA = I,   det A = 1,   eigenvalues  λ = ± i
    [ 1   0 ]                              (|λ| = 1, but complex, not ±1)

A is orthogonal and full rank, yet its eigenvalues are +i and −i — on the unit circle, neither +1 nor −1. The correct picks are orthogonal, full rank, det = ±1; the eigenvalue option is the distractor.

A question to carry forward

An orthogonal matrix preserves everything — it keeps the whole space, just rigidly turned. The opposite extreme is a matrix that throws information away on purpose: it casts a shadow, flattening every vector down onto a chosen subspace and leaving it there. Here is the thread onward: what does such a “shadow-casting” matrix look like, and what happens if you apply it twice?

In one breath

• Orthogonal Q: orthonormal columns ⇒ QᵀQ = I ⇒ Q⁻¹ = Qᵀ (inverse is free).
• Rigid: preserves length and angle — ‖Qx‖ = ‖x‖, (Qx)·(Qy) = x·y (rotations and reflections).
• Determinant: det(Q) = ±1 (+1 rotation, −1 reflection).
• Eigenvalues lie on the unit circle, |λ| = 1 — but may be complex (a rotation by θ → cosθ ± i sinθ), NOT necessarily ±1. ← the 2025 trap.
• Norm-preserving ⇒ orthogonal ⇒ full rank, det = ±1 (but not “eigenvalues ±1”).

Practice