Orthogonal matrix

Definition and Properties

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, meaning that the matrix multiplied by its transpose results in the identity matrix. Formally, a matrix \( Q \) is orthogonal if \( Q^T Q = Q Q^T = I \), where \( Q^T \) is the transpose of \( Q \) and \( I \) is the identity matrix.

Orthogonal matrices have several important properties:

The rows and columns of an orthogonal matrix are orthonormal vectors.
The determinant of an orthogonal matrix is either +1 or -1.
Orthogonal matrices preserve the dot product, meaning that for any vectors \( u \) and \( v \), \( Q(u \cdot v) = (Qu) \cdot (Qv) \).
Orthogonal matrices preserve the Euclidean norm, meaning that \( \|Qx\| = \|x\| \) for any vector \( x \).

Examples and Applications

Orthogonal matrices appear in various areas of mathematics and applied sciences. Some common examples include:

The identity matrix \( I \) is trivially orthogonal.
Rotation matrices in two and three dimensions are orthogonal.
Reflection matrices are orthogonal.

Orthogonal matrices are used in numerous applications such as:

QR decomposition, which is used in solving linear systems and eigenvalue problems.
Principal Component Analysis (PCA), which is used in statistics and machine learning for dimensionality reduction.
Computer graphics, where orthogonal matrices are used for transformations like rotations and reflections.

Construction and Decomposition

Orthogonal matrices can be constructed in several ways:

From Gram-Schmidt process, which orthogonalizes a set of vectors.
Using Householder transformations, which are used in numerical linear algebra to zero out subdiagonal elements.
From Givens rotations, which are used to introduce zeros in specific positions of a matrix.

Orthogonal matrices can also be decomposed into simpler components:

The QR decomposition expresses a matrix as the product of an orthogonal matrix \( Q \) and an upper triangular matrix \( R \).
The Singular Value Decomposition (SVD) expresses a matrix as the product of two orthogonal matrices and a diagonal matrix.

Properties in Higher Dimensions

In higher dimensions, orthogonal matrices retain their fundamental properties but exhibit more complex behaviors. For instance:

In \( n \)-dimensions, an orthogonal matrix represents a combination of rotations and reflections.
The space of all \( n \times n \) orthogonal matrices forms a group under matrix multiplication, known as the Orthogonal group \( O(n) \).

Numerical Stability and Computation

Orthogonal matrices are particularly important in numerical computations due to their stability properties:

They are used in algorithms that require numerical stability, such as the QR algorithm for eigenvalue computation.
Orthogonal transformations are used in Least Squares problems to minimize numerical errors.