Problem 13 - Algebra

Question

Zeigen Sie für \(n \in \mathbb{N}\) und jeden Körper \(K\) : (a) Die Menge \(\mathrm{O}(n, K)=\left\\{A \in K^{n \times n} \mid A A^{\top}=E_{n}\right\\}\) der orthogonalen \(n \times n\)-Matrizen bildet eine Untergruppe von \(\mathrm{GL}(n, K)\). (b) Die Menge \(\mathrm{SO}(n, K)=\\{A \in \mathrm{O}(n, K) \mid \operatorname{det}(A)=1\\}\) der speziellen orthogonalen \(n \times n\)-Matrizen bildet eine Untergruppe von \(\mathrm{O}(n, K)\).

Step-by-Step Solution

Answer

Both \( \mathrm{O}(n, K) \) and \( \mathrm{SO}(n, K) \) are subgroups of their respective groups, satisfying closure, identity, and inverses.

Step 1: Define Orthogonal Matrices

The orthogonal matrices in \( \mathrm{O}(n, K) \) are defined such that for a matrix \( A \), the relation \( AA^{\top} = E_n \) holds, where \( E_n \) is the identity matrix. This is the defining property of orthogonal matrices.

Step 2: Verify Closure under Multiplication for Orthogonal Group

Consider two matrices \( A, B \in \mathrm{O}(n, K) \). We need to show that their product, \( AB \), is also in \( \mathrm{O}(n, K) \). Since \( A \) and \( B \) are orthogonal, \( AA^{\top} = E_n \) and \( BB^{\top} = E_n \). We have:\[(AB)(AB)^{\top} = ABB^{\top}A^{\top} = AE_nA^{\top} = AA^{\top} = E_n.\]Thus, \( AB \) is orthogonal, proving closure under multiplication.

Step 3: Verify Existence of Identity for Orthogonal Group

The identity matrix \( E_n \) is trivially in \( \mathrm{O}(n, K) \) because \( E_nE_n^{\top} = E_n \). Thus, \( \mathrm{O}(n, K) \) contains the identity element.

Step 4: Verify Existence of Inverses for Orthogonal Group

For any orthogonal matrix \( A \in \mathrm{O}(n, K) \), we have \( A^{-1} = A^{\top} \). Since \( AA^{\top} = E_n \), \( A^{\top} \) is also orthogonal, and hence, \( \mathrm{O}(n, K) \) contains inverses for each of its elements.

Step 5: Define Special Orthogonal Matrices

The special orthogonal matrices in \( \mathrm{SO}(n, K) \) consist of those matrices \( A \in \mathrm{O}(n, K) \) that additionally have \( \det(A) = 1 \). This is an extra condition for being special.

Step 6: Verify Closure under Multiplication for Special Orthogonal Group

Consider two matrices \( A, B \in \mathrm{SO}(n, K) \). We need to show their product \( AB \) is in \( \mathrm{SO}(n, K) \). For orthogonal matrices, we know:\[ \det(AB) = \det(A)\det(B) = 1 \times 1 = 1. \]This confirms that \( AB \) is in \( \mathrm{SO}(n, K) \).

Step 7: Verify Existence of Identity for Special Orthogonal Group

The identity matrix \( E_n \), with \( \det(E_n) = 1 \), is in \( \mathrm{SO}(n, K) \). Thus, \( \mathrm{SO}(n, K) \) contains the identity element.

Step 8: Verify Existence of Inverses for Special Orthogonal Group

For any matrix \( A \in \mathrm{SO}(n, K) \), its inverse is \( A^{\top} \) because \( AA^{\top} = E_n \). Additionally, \( \det(A^{\top}) = \det(A) = 1 \). Thus, \( A^{\top} \) is in \( \mathrm{SO}(n, K) \), ensuring the existence of inverses.

Key Concepts

Orthogonal Matrices

Orthogonal matrices form a fascinating subset of square matrices. To belong to the set of orthogonal matrices, a matrix \(A\) should satisfy the condition \( AA^{\top} = E_n \), where \(E_n\) denotes the identity matrix of size \(n\). This condition shows that the transpose of \(A\), which is denoted as \(A^{\top}\), is also its inverse. In simpler terms:

• The rows and columns of an orthogonal matrix are orthonormal vectors, meaning they have a length of one and are at right angles (orthogonal) to each other.
• When you multiply an orthogonal matrix by its transpose, you end up with the identity matrix.
• The determinant of an orthogonal matrix is either \(1\) or \(-1\).

These properties make orthogonal matrices useful in a variety of applications, such as computer graphics and signal processing, where preserving vector norms and angles under transformation is crucial.

Special Orthogonal Group

The special orthogonal group, often denoted as \(\mathrm{SO}(n, K)\), is a subgroup of the orthogonal group \(\mathrm{O}(n, K)\). Matrices in the special orthogonal group have all the properties of orthogonal matrices with an additional criterion: their determinant is +1. This determinant condition is crucial because:

• Determinants provide information about the matrix's scaling factor upon transformation. A determinant of 1 indicates volume-preserving transformations.
• The special orthogonal group contains rotation matrices, which are central to understanding rigid body movements in physics.

By requiring the determinant to be 1, \(\mathrm{SO}(n, K)\) focuses on transformations that preserve orientation and don't involve reflection, making them essential for disciplines like robotics and 3D modeling.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, but it follows a different set of rules than everyday algebra. When you multiply two matrices \(A\) and \(B\), the resulting matrix has dimensions determined by the number of rows in \(A\) and columns in \(B\). Here are key points about matrix multiplication:

• The operation is not commutative, meaning \(AB\) does not necessarily equal \(BA\).
• It involves a series of dot products of the rows of \(A\) with the columns of \(B\).
• Orthogonal and special orthogonal matrices, as seen before, are closed under this operation, meaning their product remains within the same matrix group.

Understanding matrix multiplication is not only vital for theory but also practical for computer algorithms like 3D graphics transformations and machine learning models.

Identity Matrix

The identity matrix, denoted as \(E_n\) for a square matrix of size \(n\times n\), acts as a multiplicative identity in the space of matrices, similar to how the number 1 is the identity for multiplication among numbers. Its unique feature is that it leaves other matrices unchanged when used in multiplication. Consider the properties of the identity matrix:

• All the diagonal elements are 1, while all off-diagonal elements are 0.
• If \(A\) is any matrix and \(E_n\) is the identity matrix, then \(AE_n = A\) and \(E_nA = A\).
• The identity matrix plays a crucial role in defining orthogonal and special orthogonal matrices, as they must satisfy the condition involving their transpose or inverse, resulting in the identity matrix.

It serves as a foundational building block for more complex linear algebraic concepts, making it essential for both academic exploration and practical applications.