orthogonal matrix

The determinant of any orthogonal matrix is either +1 or 1.

A general orthogonal matrix has only one real eigenvalue, either +1 or 1.

Orthogonal matrices are important for a number of reasons, both theoretical and practical.

The determinant of a rotation orthogonal matrix must be 1.

Below are a few examples of small orthogonal matrices and possible interpretations.

In this case, because and are real valued, they each are an orthogonal matrix.

Orthogonal matrices are of very great importance in dynamics.

The product of two such matrices is a special orthogonal matrix which represents a rotation.

As it is an orthogonal matrix these diagonal elements are either 1 or 1.

A subtle technical problem afflicts some uses of orthogonal matrices.

Matrices for which this property holds are called orthogonal matrices.

Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant.

Then, any orthogonal matrix is either a rotation or an improper rotation.

More generally, coordinate rotations in any dimension are represented by orthogonal matrices.

As a linear transformation, every special orthogonal matrix acts as a rotation.

In the language of matrix (mathematics), rotations are special orthogonal matrix.

So to first order, an infinitesimal rotation matrix is an orthogonal matrix.

Since a trace is invariant under an orthogonal matrix transformation:

The real analogue of a unitary matrix is an orthogonal matrix.

These groups are readily constructed with two-dimensional orthogonal matrices.

A is a real, orthogonal matrix, hence each of its rows or columns represents a unit vector.

More specifically they can be characterized as orthogonal matrices with determinant 1:

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix.

Every orthogonal matrix has determinant 1 or 1.

A number of important matrix decompositions involve orthogonal matrices, including especially: