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Orthogonal Matrices

Orthogonal-matrix checker. Enter A to test whether AᵀA=I, with residual—for orthogonality practice.

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AᵀA = I ⇔ A⁻¹ = Aᵀ
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜Orthogonal Matrices

1. Definition
A square matrix with AᵀA = I is orthogonal, so A⁻¹ = Aᵀ.
2. Symbols
SymbolMeaning
Iidentity matrix
A⁻¹inverse
Aᵀtranspose
3. How this tool works
  • Columns are unit vectors and pairwise orthogonal;
  • |det A| = 1;
  • Orthogonal transformations preserve lengths and angles.
4. Steps
  1. Enter A;
  2. Click Calculate to compute AᵀA;
  3. Check whether it equals I.
5. Example
Example: A=0 1;−1 0. Aᵀ=0 −1;1 0, AᵀA=1 0;0 1=I, so it is orthogonal.
6. Pitfalls
Must be square;
A near-identity within 1e-6 is treated as orthogonal;
Rows are also orthonormal.

❓ FAQ｜Orthogonal Matrices

What is an orthogonal matrix?
A square matrix satisfying AᵀA=I, i.e. A⁻¹=Aᵀ.
What about its columns?
Columns are unit vectors and pairwise orthogonal.
Determinant of an orthogonal matrix?
|det A|=1 (+1 for rotation, −1 for reflection).
Geometric meaning?
Preserves vector lengths and angles between vectors.
Tolerance?
AᵀA within 1e-6 of I is treated as orthogonal.