Properties of Orthogonal Matrices

Orthogonal matrices are an essential concept in linear algebra with various properties that make them uniquely useful in many mathematical and computational applications. In this note, we will explore the definition of orthogonal matrices, their key properties, and their significance in mathematics.

Definition

An orthogonal matrix is a square matrix $Q$ such that $Q^{T}Q=I$, where $Q^T$ denotes the transpose of $Q$ and $I$ is the identity matrix. In simple terms, the columns of an orthogonal matrix are orthogonal unit vectors, and the dot product of any two columns is zero unless they are the same column.

Properties

1. Orthogonality: The defining property of an orthogonal matrix is that $Q^{T}Q=I$.
2. Preservation of Length: Orthogonal matrices preserve the length of vectors. For any vector $v$, $||Qv||=||v||$, where $||\cdot ||$ denotes the Euclidean norm.
3. Preservation of Dot Product: For any vectors $v$ and $w$, the dot product is preserved under multiplication by an orthogonal matrix: $(Qv{)}^T(Qw)=v^{T}w$.
4. Determinant: The determinant of an orthogonal matrix is either $1$ or $-1$.
5. Inverse: The inverse of an orthogonal matrix is its transpose, i.e., $Q^{-1}=Q^T$.

Significance

Orthogonal matrices play a crucial role in many areas of mathematics and its applications. They are extensively used in rotations, reflections, and transformations in geometry, computer graphics, physics, and signal processing. The preservation of lengths and angles under multiplication by orthogonal matrices makes them particularly valuable in these fields.

Historically, the concept of orthogonality dates back to Ancient Greece, where Euclid studied geometric properties of shapes and angles. The modern definition of orthogonal matrices emerged in the 19th century with the development of linear algebra by mathematicians like Carl Friedrich Gauss and Augustin-Louis Cauchy.

Examples

Let’s consider an example of a $2\times 2$ orthogonal matrix:

Q=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}$$ where $\theta$ is any angle. This matrix represents a 2D rotation by angle $\theta$. ## Exam Questions 1. Explain the properties of orthogonal matrices and give an example of a $3\times 3$ orthogonal matrix. 2. Prove that the determinant of an orthogonal matrix is either $1$ or $-1$. 3. Show that the columns of an orthogonal matrix form an orthonormal basis in $\mathbb{R}^n$. Remember that practice is key to mastering any mathematical concept!