Details and Options
- A p×q matrix m is orthogonal if p≥q and Transpose[m].m is the q×q identity matrix, or p≤q and m.Transpose[m] is the p×p identity matrix.
- OrthogonalMatrixQ works for symbolic as well as numerical matrices.
- The following options can be given:
Normalized True test if matrix rows are normalized SameTest Automatic function to test equality of expressions Tolerance Automatic tolerance for approximate numbers
- For exact and symbolic matrices, the option SameTest->f indicates that two entries aij and bij are taken to be equal if f[aij,bij] gives True.
- For approximate matrices, the option Tolerance->t can be used to indicate that the norm γ=m.mT-In∞ satisfying γ≤t is taken to be zero where In is the identity matrix.
Examplesopen allclose all
An approximate MachinePrecision matrix:
Generalizations & Extensions (1)
Orthogonalize applied to real vectors generates an orthogonal matrix:
Check that a matrix drawn from CircularRealMatrixDistribution is orthogonal:
Properties & Relations (10)
Dot products of orthogonal matrices are orthogonal:
The matrix exponential MatrixExp of an antisymmetric matrix is always orthogonal:
Wolfram Research (2014), OrthogonalMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/OrthogonalMatrixQ.html.
Wolfram Language. 2014. "OrthogonalMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/OrthogonalMatrixQ.html.
Wolfram Language. (2014). OrthogonalMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/OrthogonalMatrixQ.html