Orthogonal matrix

A real square matrix whose transpose is its inverse. Its columns and rows form orthonormal bases, so it preserves Euclidean lengths and angles.

Orthonormal basis: A basis whose vectors are mutually perpendicular and each have unit length. The columns of an orthogonal matrix form an orthonormal basis.
Rotation matrix: A matrix representing a rotation as a linear transformation of coordinates. Proper rotations are orthogonal matrices with determinant 1.
Unitary matrix: A complex square matrix whose conjugate transpose is its inverse. It generalizes orthogonal matrices from real entries to complex entries.
Transpose: The matrix formed by exchanging a matrix's rows and columns. The transpose appears directly in the defining equation QᵀQ = I.
Euclidean norm: The length of a vector defined by the square root of the sum of its squared coordinates. Orthogonal matrices preserve this length for every vector.
Householder transformation: An orthogonal reflection across a hyperplane, commonly represented by a matrix. Householder matrices reduce matrices to simpler forms while preserving norms.
Symmetric matrix: A square matrix equal to its transpose. Symmetry constrains a matrix differently from the inverse-transpose condition.
Inverse matrix: A square matrix that reverses the transformation represented by another matrix. An orthogonal matrix is characterized by having its transpose as its inverse.
Inner product: A bilinear operation that generalizes the dot product and encodes lengths and angles. Preserving inner products is equivalent to preserving lengths and angles.
QR decomposition: A matrix factorization expressing a matrix as a product of an orthogonal matrix and an upper triangular matrix. Its orthogonal factor supports stable solutions of linear systems.

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