Mathematics.

matrix theory

Unitary and Orthogonal Matrices

Linear Algebra40 minDifficulty6 out of 10

You should know: inner product spaces, matrices

Overview

An orthogonal matrix (real) or unitary matrix (complex) is a square matrix whose columns form an orthonormal set — equivalently, a matrix that preserves lengths and angles. These matrices represent isometries: rotations, reflections, or their compositions. They are the cornerstone of QR decomposition, Schur decomposition, and SVD.

Intuition

Multiplying by an orthogonal/unitary matrix is like rotating or reflecting space — it moves vectors around without stretching or squishing them. Since lengths are preserved, these transformations are 'rigid motions' of the vector space. The condition Q^T Q = I (or Q* Q = I for complex) means the columns of Q are perpendicular unit vectors, which is the cleanest possible basis.

Formal Definition

Definition

A real square matrix Q is orthogonal if Q^T Q = I. A complex square matrix U is unitary if U*U = I (where U* is the conjugate transpose).

Q⊤Q=I  ⟺  Q⊤=Q−1(orthogonal)Q^\top Q = I \iff Q^\top = Q^{-1} \quad (\text{orthogonal})
Orthogonal matrix
U∗U=I  ⟺  U∗=U−1(unitary)U^* U = I \iff U^* = U^{-1} \quad (\text{unitary})
Unitary matrix
∣det⁡(Q)∣=1(so det⁡(Q)=±1 for real, ∣det⁡(U)∣=1 for complex)|\det(Q)| = 1 \quad (\text{so } \det(Q) = \pm 1 \text{ for real, } |\det(U)|=1 \text{ for complex})
Determinant constraint

Notation

NotationMeaning
O(n)O(n)Group of n×n real orthogonal matrices
SO(n)SO(n)Special orthogonal group: det = +1 (rotations only)
U(n)U(n)Group of n×n unitary matrices

Properties

Length-preserving

∥Qv∥=∥v∥ for all v.\|Qv\| = \|v\| \text{ for all } v.

Angle-preserving

⟨Qu,Qv⟩=⟨u,v⟩ for all u,v.\langle Qu, Qv\rangle = \langle u,v\rangle \text{ for all } u,v.

Eigenvalues on unit circle

All eigenvalues of a unitary matrix satisfy ∣λ∣=1.\text{All eigenvalues of a unitary matrix satisfy } |\lambda|=1.

Group closure

Products and inverses of orthogonal/unitary matrices are orthogonal/unitary.\text{Products and inverses of orthogonal/unitary matrices are orthogonal/unitary.}

Worked Examples

  1. 1

    Compute Q^T Q.

    Q⊤=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)Q^\top = \begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix}
  2. 2

    Multiply Q^T by Q.

    Q⊤Q=(cos⁡2θ+sin⁡2θ−cos⁡θsin⁡θ+sin⁡θcos⁡θsin⁡θcos⁡θ−cos⁡θsin⁡θsin⁡2θ+cos⁡2θ)=(1001)=IQ^\top 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

✓ Answer

Q^T Q = I, so Q is orthogonal. It represents a rotation by angle θ, with det(Q)=cos²θ+sin²θ=1.

Practice Problems

Mediumfree response

Is A = [[1,1],[0,1]] orthogonal? Justify.

Mediumfree response

For U = (1/2)[[1+i, 1-i],[1-i, 1+i]], verify U is unitary.

Common Mistakes

Common Mistake

Thinking Q^T = Q^{-1} means Q is symmetric.

Q^T = Q^{-1} is the definition of orthogonality; a symmetric matrix satisfies Q = Q^T, which is different unless Q is also the identity.

Quiz

A real matrix Q is orthogonal if and only if:
Multiplying a vector by an orthogonal matrix:
The eigenvalues of a unitary matrix all satisfy:

Summary

  • A real matrix Q is orthogonal if Q^T Q = I; a complex matrix U is unitary if U*U = I.
  • Orthogonal/unitary matrices preserve lengths (||Qv|| = ||v||) and inner products.
  • Determinant of an orthogonal matrix is ±1; of a unitary matrix has absolute value 1.
  • Eigenvalues of a unitary matrix all lie on the complex unit circle.
  • Orthogonal/unitary matrices form groups under multiplication: O(n), SO(n), U(n).

References

  1. WebsiteWikipedia — Unitary matrix