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Chapter 10: Problem 17

Zeigen Sie, dass \(O(n)\) und \(\mathrm{SO}(n)\) (mit der Matrizenmultiplikation als Verkn眉pfung) Gruppen sind.

Short Answer

Expert verified
Both \(O(n)\) and \(\mathrm{SO}(n)\) satisfy group properties: closure, identity, inverses, and associativity.

Step by step solution

01

Define the Groups

Define the orthogonal group \(O(n)\) as the set of all \(n \times n\) orthogonal matrices \(A\) such that \(A^T A = I\), where \(A^T\) is the transpose of \(A\) and \(I\) is the identity matrix. Define the special orthogonal group \(\mathrm{SO}(n)\) as the subset of \(O(n)\) consisting of matrices with determinant 1.
02

Show Closure under Multiplication

Both \(O(n)\) and \(\mathrm{SO}(n)\) are closed under matrix multiplication. For \(O(n)\), if \(A, B \in O(n)\), then \((AB)^T (AB) = B^T A^T A B = B^T I B = B^T B = I\). For \(\mathrm{SO}(n)\), the additional requirement of \(\det(AB) = \det(A)\det(B) = 1 \times 1 = 1\) holds, hence \(AB \in \mathrm{SO}(n)\).
03

Identity Element

The identity matrix \(I\) satisfies \(I^T I = I\) and \(\det(I) = 1\), so it is in both \(O(n)\) and \(\mathrm{SO}(n)\). It acts as the identity element for the matrix multiplication operation because \(AI = IA = A\) for any matrix \(A\) in the groups.
04

Existence of Inverses

For any matrix \(A \in O(n)\), the inverse matrix \(A^{-1}\) exists and is equal to \(A^T\), because \(A^T A = I\) implies \(A A^T = I\). For \(\mathrm{SO}(n)\), since \(\det(A) = 1\), \(\det(A^T) = 1\) and \( (A^T)^T A^T = I\), thus inv(A) is also in \(\mathrm{SO}(n)\).
05

Associativity Property

The associativity of matrix multiplication is a well-known property: for any matrices \(A, B, C\) that are appropriately sized, \((AB)C = A(BC)\). Therefore, matrix multiplication is associative for both \(O(n)\) and \(\mathrm{SO}(n)\).
06

Conclusion

Since all group properties (closure, identity, inverses, and associativity) are satisfied, both \(O(n)\) and \(\mathrm{SO}(n)\) are groups under matrix multiplication.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Orthogonal Group
The orthogonal group, denoted as \( O(n) \), is a collection of \( n \times n \) matrices with a unique property. These matrices are called orthogonal because when you multiply the matrix by its transpose, the result is the identity matrix. This is expressed as \( A^T A = I \). Here, \( A^T \) stands for the transpose of matrix \( A \), and \( I \) represents the identity matrix, looking much like the number 1, just in matrix form.
Orthogonal matrices are special because:
  • They preserve the lengths of vectors during transformations.
  • They maintain the angles between vectors.
  • These matrices are significant in rigid transformations, often used in geometry.
They are an essential building block in linear algebra, showcasing symmetry and conservation properties.
Special Orthogonal Group
The special orthogonal group, typically written as \( \mathrm{SO}(n) \), consists of matrices that belong to the orthogonal group \( O(n) \), but with an added twist. This twist is that each matrix in \( \mathrm{SO}(n) \) has a determinant equal to 1.
Determinants are like fingerprints for matrices, and a matrix belonging to \( \mathrm{SO}(n) \) still maintains the orthogonal property of preserving vector norms and angles. Still, it must have this particular determinant value.
Special orthogonal matrices are important because:
  • They represent transformations that are rotations without any reflection.
  • They maintain orientation, which means they don't flip left to right.
  • These matrices are heavily used in physics and engineering to describe rotational dynamics.
Therefore, these matrices are not only orthogonal but also have the special feature of preserving orientation.
Matrix Multiplication
Matrix multiplication is the process of multiplying two matrices together to form a new matrix. It involves combining rows from the first matrix with columns of the second matrix. This operation is a cornerstone of matrix algebra and plays a crucial role in describing transformations, among other applications.
Key points to note about matrix multiplication:
  • The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be valid.
  • The resulting matrix will have dimensions that reflect the rows of the first matrix and the columns of the second matrix.
  • Matrix multiplication is associative but not commutative, meaning \( (AB)C = A(BC) \) but not necessarily \( AB = BA \).
In the context of groups like \( O(n) \) and \( \mathrm{SO}(n) \), matrix multiplication is used to show that these groups are closed under the operation鈥攎eaning multiplying any two matrices from within the group will result in another matrix that is also part of the group.
Group Properties
In mathematics, a group refers to a set equipped with an operation meeting certain criteria. This structure is foundational to various fields, including linear algebra. For a set to qualify as a group, it must satisfy the following properties:
  • Closure: Performing the group operation on any two elements within the group results in another element also in the group.
  • Associativity: The order in which the operation is performed doesn't change the result, as long as the sequence of elements is the same, illustrated by \( (AB)C = A(BC) \).
  • Identity Element: There exists an element in the group that, when used in the operation with any other group element, leaves the other element unchanged.
  • Inverse: Every element in the group has an inverse, a special element that combines with the original to result in the identity element.
The orthogonal group \( O(n) \) and the special orthogonal group \( \mathrm{SO}(n) \) satisfy all these properties, hence they are both considered groups under matrix multiplication.
Identity Matrix
The identity matrix, symbolized as \( I \), is a special type of matrix in the world of linear algebra. It acts much like the number 1 in regular arithmetic, where multiplying any number by 1 leaves it unchanged. Similarly, multiply any matrix by an identity matrix, and the original matrix remains the same.
Key characteristics of the identity matrix include:
  • It is a square matrix, having the same number of rows and columns.
  • It has 1s on its main diagonal (stretching from the top left to the bottom right) and 0s everywhere else.
  • The identity matrix satisfies the equation \( AI = IA = A \), where \( A \) is any \( n \times n \) matrix.
Within groups like \( O(n) \) and \( \mathrm{SO}(n) \), the identity matrix exerts its presence by serving as the neutral element crucial for the inverse and providing a foundation for group operations.

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Most popular questions from this chapter

Im ersten Teil des Beweises f眉r den Orthonormalisierungssatz haben wir die F盲lle \(\left\langle v_{2}, w_{1}\right\rangle=0\) und \(\left\langle v_{2}, w_{1}\right\rangle \neq 0\) unterschieden. War das notwendig?

Geben Sie eine Gram-Matrix eines Skalarprodukts eines 4 -dimensionalen reellen Vektorraums an, deren s盲mtliche Eintr盲ge \(\neq 0\) sind.

Sei \(B=\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\) eine Basis des euklidischen Vektorraums \(V\). Zeigen Sie: Eine lineare Abbildung von \(V\) in sich ist genau dann selbstadjungiert, wenn \(\left\langle f\left(v_{i}\right), v_{j}\right\rangle=\left\langle v_{i}, f\left(v_{j}\right)\right\rangle\) f眉r alle \(i, j \in\\{1,2, \ldots, n\\}\) gilt.

Zeigen Sie: Die Menge der symmetrischen Matrizen aus \(K^{n \times n}\) bildet einen \(K\) Vektorraum. Bestimmen sie die Dimension dieses Vektorraums.

Zeigen Sie: Ist \(\psi\) eine selbstadjungierte lineare Abbildung von \(V\) in sich, so wird durch $$ [v, w]:=\langle\psi(v), w\rangle $$ eine symmetrische Bilinearform definiert.
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