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 columns 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
Basic Examples (2)
Basic Uses (6)
Use OrthogonalMatrixQ with arbitrary-precision matrix:
Use OrthogonalMatrixQ with a symbolic matrix:
OrthogonalMatrixQ works efficiently with large numerical matrices:
Special Matrices (4)
Rectangular Semi-orthogonal Matrices (4)
However, it will not give true for Transpose[m]:
Sources of Orthgonal Matrices (5)
Orthogonalize applied to real, linearly independent vectors generates an orthogonal matrix:
Matrices drawn from CircularRealMatrixDistribution are orthogonal:
Uses of Orthogonal Matrices (5)
Any orthogonal matrix represents a rotation and/or reflection. If the matrix has determinant , it is a pure rotation. If it the determinant is , the matrix includes a reflection. Consider the following matrix:
Properties & Relations (13)
Use Eigenvalues to find eigenvalues:
Use Eigenvectors to find eigenvectors:
MatrixExp[m] for real antisymmetric m is both orthogonal and unitary:
Possible Issues (1)
OrthogonalMatrixQ uses the definition for both real- and complex-valued matrices:
UnitaryMatrixQ tests the more common definition that ensures a complex matrix is normal:
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