Orthogonal Matrix Checker
Check whether a square matrix satisfies A·Aᵀ = the identity matrix.
Check whether a square matrix satisfies A·Aᵀ = the identity matrix.
How Orthogonal Matrix Checker Works
The calculator multiplies the matrix by its transpose and checks whether the result is the identity matrix — when it is, the matrix's transpose is also its inverse, and the matrix represents a pure rotation or reflection that preserves length and angles.
See It In Action
Who Uses Orthogonal Matrix Checker and Why
- Verifying a rotation matrix used in computer graphics or robotics preserves lengths and angles without introducing distortion.
- Checking a change-of-basis matrix for an orthonormal basis before relying on that property in a linear algebra calculation.
- Confirming a camera or object transform in a 3D application doesn't unintentionally stretch or skew the scene.
Mistakes to Avoid
- Assuming a matrix with rows that look roughly perpendicular is automatically orthogonal — the rows must also each have length exactly 1 (be orthonormal), not just be at right angles to each other.
- Testing a non-square matrix — the identity-matrix comparison A·Aᵀ = I this check performs only makes sense for square matrices.
- Confusing this specific A·Aᵀ = I test with the more general, looser idea of "having orthogonal rows," which doesn't require unit length.
Tips for Best Results
- A quick sanity check: an orthogonal matrix always has determinant exactly +1 (a pure rotation) or −1 (a reflection, or a rotation combined with one).
- Because orthogonal matrices preserve distance and angle, they're the standard tool for representing rotations in graphics, robotics, and physics without introducing scaling artifacts.
Fixing Common Problems
My rows look perpendicular to me but the check failed. — Check the length of each row vector — perpendicular rows that aren't also unit length (magnitude 1) will fail this specific test even though they'd pass a looser "orthogonal rows" check.
Terms Explained
Orthogonal matrix: A square matrix whose transpose is also its inverse (A·Aᵀ = the identity matrix), representing a rotation or reflection that preserves lengths and angles.
Orthonormal: A set of vectors that are both mutually perpendicular (orthogonal) and each of unit length.