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Orthogonal Matrix — Definition, Formula & Examples

An orthogonal matrix is a square matrix whose columns (and rows) are orthonormal vectors, which means its transpose is also its inverse. Multiplying an orthogonal matrix by its transpose gives the identity matrix.

A real n×nn \times n matrix QQ is orthogonal if and only if QTQ=QQT=InQ^T Q = Q Q^T = I_n, where QTQ^T denotes the transpose and InI_n is the n×nn \times n identity matrix. Equivalently, Q−1=QTQ^{-1} = Q^T, and det⁡(Q)=±1\det(Q) = \pm 1.

Key Formula

QTQ=I⟺Q−1=QTQ^T Q = I \quad \Longleftrightarrow \quad Q^{-1} = Q^T
Where:
  • QQ = An $n \times n$ real matrix
  • QTQ^T = The transpose of $Q$
  • II = The $n \times n$ identity matrix

How It Works

To check whether a matrix is orthogonal, compute QTQQ^T Q and see if you get the identity matrix. This amounts to verifying two things: each column has length 1, and every pair of distinct columns has a dot product of 0. Because the inverse of an orthogonal matrix is simply its transpose, solving systems or reversing transformations becomes computationally cheap. Orthogonal matrices represent geometric operations — rotations and reflections — that preserve lengths and angles.

Worked Example

Problem: Determine whether the matrix Q=(0−110)Q = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} is orthogonal.
Step 1: Compute the transpose of QQ.
QT=(01−10)Q^T = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
Step 2: Multiply QTQQ^T Q and check if the result is the identity matrix.
QTQ=(01−10)(0−110)=(1001)=IQ^T Q = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I
Step 3: Verify the determinant as a quick sanity check.
det⁡(Q)=(0)(0)−(−1)(1)=1\det(Q) = (0)(0) - (-1)(1) = 1
Answer: Since QTQ=IQ^T Q = I and det⁡(Q)=1\det(Q) = 1, the matrix QQ is orthogonal. It represents a 90° counterclockwise rotation.

Why It Matters

Orthogonal matrices are central to the QR decomposition, singular value decomposition (SVD), and principal component analysis (PCA). In computer graphics, they encode rotations of 3D objects efficiently. Their numerical stability — they never amplify rounding errors — makes them essential in scientific computing.

Common Mistakes

Mistake: Assuming any matrix with orthogonal columns is an orthogonal matrix.
Correction: The columns must be orthonormal — both mutually perpendicular and each of unit length. A matrix with orthogonal but non-unit columns satisfies QTQ=DQ^T Q = D (a diagonal matrix), not QTQ=IQ^T Q = I.