OrthogonalMatrixQ
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 a_{ij} and b_{ij} are taken to be equal if f[a_{ij},b_{ij}] gives True.
 For approximate matrices, the option Tolerance>t can be used to indicate that the norm γ=m.m^{T}I_{n}_{∞} satisfying γ≤t is taken to be zero where I_{n} is the identity matrix.
Examples
open allclose allBasic Examples (2)
Scope (14)
Basic Uses (6)
Test if a real matrix is orthogonal:
A real orthogonal matrix is also unitary:
Test if a complex matrix is orthogonal:
A complexvalued orthogonal matrix is not unitary:
Test if an exact matrix is orthogonal:
Use OrthogonalMatrixQ with arbitraryprecision matrix:
A random matrix is typically not orthogonal:
Use OrthogonalMatrixQ with a symbolic matrix:
The matrix becomes orthogonal when and :
OrthogonalMatrixQ works efficiently with large numerical matrices:
Special Matrices (4)
Use OrthogonalMatrixQ with sparse matrices:
Use OrthogonalMatrixQ with structured matrices:
The identity matrix is orthogonal:
HilbertMatrix is not orthogonal:
Rectangular Semiorthogonal Matrices (4)
Test if a rectangular matrix is semiorthogonal:
As there are more columns than rows, this indicates that the rows are orthonormal:
The columns are not orthonormal:
Test a matrix with more rows than columns:
The columns of the matrix are orthonormal:
Generate a random orthogonal matrix:
Any subset of its rows forms a rectangular semiorthogonal matrix:
Options (4)
Normalized (2)
Symbolic orthogonal matrix columns are often not normalized to 1:
Avoid testing if the columns are normalized:
Multiply the second column of an orthogonal matrix by 2:
OrthogonalMatrixQ with NormalizedFalse will still give True for m:
However, it will not give true for Transpose[m]:
SameTest (1)
This matrix is orthogonal for a positive real , but OrthogonalMatrixQ gives False:
Use the option SameTest to get the correct answer:
Applications (10)
Sources of Orthogonal Matrices (5)
Any orthonormal basis for forms an orthogonal matrix:
Putting the basis vectors in rows of a matrix forms an orthogonal matrix:
Putting them in columns also gives an orthogonal matrix:
Orthogonalize applied to real, linearly independent vectors generates an orthogonal matrix:
The matrix does not need to be square, in which case the resulting matrix is semiorthogonal:
But the starting matrix must have full rank:
Any rotation matrix is orthogonal:
Any permutation matrix is orthogonal:
Matrices drawn from CircularRealMatrixDistribution are orthogonal:
Uses of Orthogonal Matrices (5)
Orthogonal matrices preserve the standard inner product on . In other words, if is orthogonal and and are vectors, then :
This means the angles between the vectors are unchanged:
Since the norm is derived from the inner product, norms are preserved as well:
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:
It is orthogonal and has determinant :
Thus, it is a pure rotation; the Cartesian unit vectors and maintain their relative positions:
The following matrix is orthogonal but has determinant :
Thus, it includes a reflection; the Cartesian unit vectors and reverse their relative positions:
Orthogonal matrices play an important role in many matrix decompositions:
The matrix is always orthogonal for any nonzero real vector :
is called a Householder reflection; as a reflection, its determinant is :
It represents a reflection through a plane perpendicular to , sending to :
Any vector perpendicular to is unchanged by :
In matrix computations, is used to set to zero selected components of a given column vector :
Find the function satisfying the following differential equation:
Represent the crossproduct with by means of multiplication by the antisymmetric matrix :
Compute the exponential and use it to define a solution to the equation:
Verify that satisfies the differential equation and initial condition:
The matrix is orthogonal for all values of :
Thus, the orbit of the solution is at a constant distance from the origin, in this case a circle:
Properties & Relations (14)
A matrix is orthogonal if m.Transpose[m]IdentityMatrix[n]:
For an approximate matrix, the identity is approximately true:
The inverse of an orthogonal matrix is its transpose:
Thus, the inverse and transpose are orthogonal matrices as well:
A real orthogonal matrix preserves the standard inner product of vectors in :
As a consequence, real orthogonal matrices preserve norms as well:
Any realvalued orthogonal matrix is unitary:
But a complex unitary matrix is typically not orthogonal:
Products of orthogonal matrices are orthogonal:
A realvalued orthogonal matrix is normal:
A complexvalued orthogonal matrix need not be normal:
Realvalued orthogonal matrices have eigenvalues that lie on the unit circle:
Use Eigenvalues to find eigenvalues:
Verify they lie on the unit circle:
This does not apply to complexvalued orthogonal matrices:
Real orthogonal matrices have a complete set of eigenvectors:
As a consequence, they must be diagonalizable:
Use Eigenvectors to find eigenvectors:
A complex orthogonal matrix can fail to be diagonalizable:
The singular values are all 1 for a real orthogonal matrix:
This need not be true for a complex orthogonal matrix:
The determinant of an orthogonal matrix is 1 or :
The 2norm of a real orthogonal matrix is always 1:
This need not be true for complex orthogonal matrices:
Integer powers of orthogonal matrices are orthogonal:
MatrixExp[m] for real antisymmetric m is both orthogonal and unitary:
For complex antisymmetric m, the exponential is orthogonal but not, in general, unitary:
OrthogonalMatrix can be used to explicitly construct orthogonal matrices:
These satisfy OrthogonalMatrixQ:
Possible Issues (1)
OrthogonalMatrixQ uses the definition for both real and complexvalued matrices:
These complex matrices need not be normal or possess many properties of real orthogonal matrices:
UnitaryMatrixQ tests the more common definition that ensures a complex matrix is normal:
Alternatively, test if the entries are real to restrict to real orthogonal matrices:
Text
Wolfram Research (2014), OrthogonalMatrixQ, Wolfram Language function, https://reference.wolfram.com/language/ref/OrthogonalMatrixQ.html.
CMS
Wolfram Language. 2014. "OrthogonalMatrixQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/OrthogonalMatrixQ.html.
APA
Wolfram Language. (2014). OrthogonalMatrixQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/OrthogonalMatrixQ.html