Construct an orthogonal matrix:

Show the elements:

Normal can convert an OrthogonalMatrix to its ordinary representation:

Matrices corresponding to isometric transformations (reflections and rotations) are orthogonal:

Real reflection matrices are both orthogonal and symmetric:

A complex orthogonal matrix:

The matrix is orthogonal, but not unitary:

A rectangular orthogonal matrix:

Its transpose is also orthogonal:

OrthogonalMatrix objects include properties that give information about the matrix:

The "Summary" property gives a brief summary of information about the matrix:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

Create random orthogonal matrices using CircularRealMatrixDistribution:

Create a wide orthogonal matrix by selecting rows from a square matrix:

Create another wide orthogonal matrix by selecting rows from this wide matrix:

Create a tall orthogonal matrix by extracting columns from a square matrix:

Create another tall orthogonal matrix by selecting rows from this tall matrix:

Matrices drawn from CircularRealMatrixDistribution are orthogonal:

A rotation matrix:

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 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 :

The transpose of a square OrthogonalMatrix is equivalent to the inverse of the original matrix:

In other words, :

Consequently, the transpose of a square orthogonal matrix is also orthogonal:

An orthogonal matrix obeys PseudoInverse[o]==ConjugateTranspose[o]:

Unlike the relationship to Inverse, this is true irrespective of shape:

For a tall orthogonal matrix , the columns are orthonormal, i.e. for :

For a wide orthogonal matrix, it is instead the rows that are orthonormal:

A matrix of dimensions is orthogonal if and is the identity matrix:

Note that is well defined, but not an identity matrix as :

If , then is orthogonal if is the identity matrix:

Again, the product in the opposite order is well defined but not an identity matrix:

The product of two square orthogonal matrices is another orthogonal matrix:

A real orthogonal matrix is also a unitary matrix:

Convert from one type of matrix to the other:

PolarDecomposition by default returns an OrthogonalMatrix object when the input is real:

OrderedSchurDecomposition by default returns an OrthogonalMatrix object when the input is real: