Orthogonal matrix

An orthogonal matrix is a square matrix $Q$ whose columns form an orthonormal basis for the vector space. This means that the columns are mutually perpendicular and have a length of $1$. In symbolic terms, for an orthogonal matrix $Q$, the following conditions hold:

• Transpose inverse property: $Q^T Q = Q Q^T = I$, where $I$ is the identity matrix.

• Orthogonality property: The dot product of any two distinct columns (or rows) of $Q$ is zero, and the dot product of a column (or row) with itself is $1$.

Example

A $2\times 2$ orthogonal matrix:

To confirm its orthogonality:

Both products result in the identity matrix $I$. Therefore, the matrix is indeed orthogonal.

Inverse of orthogonal matrix

The inverse of an orthogonal matrix $Q$ is its transpose, denoted as $Q^T$. This property stems from the fact that the transpose inverse property holds. Mathematically,

Determinant of orthogonal matrix

The determinant of an orthogonal matrix $Q$ can only take two possible values: $1$ or $-1$. This follows from the property that the columns (or rows) are unit vectors and thus do not affect volume scaling during transformations. Therefore,

Dot product of orthogonal matrix

The orthogonality property determines the dot product of any two columns (or rows) of an orthogonal matrix $Q$. If the columns (or rows) are distinct, their dot product is $0$, ensuring perpendicularity. When the columns (or rows) are the same, their dot product is $1$, reflecting the unit vector nature.

Properties

• Orthogonal matrices are attainable only from square matrices.

• All elements within an orthogonal matrix are real numbers.

• All identity matrices are orthogonal matrices.

• The product of two orthogonal matrices is also an orthogonal matrix.

• If matrix $Q$ is orthogonal, then its transpose $Q^T$ and inverse $Q^{-1}$ are also orthogonal matrices.

Applications

• Geometry: Orthogonal matrices preserve lengths and angles during transformations. This property is crucial in graphics, where objects need to be transformed without distortion.

• QR decomposition: In the QR decomposition, a matrix $A$ is decomposed into an orthogonal matrix $Q$ and an upper triangular matrix $R$. This technique finds application in solving linear equations and eigenvalue computations.

Conclusion

Orthogonal matrices in linear algebra offer the understanding of transformations, preservation of geometric properties, and mathematical operations. Their unique properties, such as the transpose inverse and orthogonality properties, make them essential in various fields, from computer graphics to physics simulations.