Question

Prove that if the operator \(A\) in a Euclidean space is skew-symmetric, then the operator \(e^{A}\) is orthogonal.

Short answer

Question: Prove that if the operator \(A\) in a Euclidean space is skew-symmetric, then the operator \(e^{A}\) is orthogonal. Answer: We should show that \((e^A)^*(e^A)=I\). We found that \(e^{A^*}=e^{-A}\), and using this, we got \((e^A)^*(e^A) = e^{-A}e^A = e^0 = I\). Since \((e^A)^*(e^A) = I\), the operator \(e^A\) is orthogonal.

Step-by-step solution

Recall the definitions of skew-symmetric and orthogonal operators

A linear operator \(A\) is called skew-symmetric if it satisfies the condition \[A = -A^*\] where \(A^*\) denotes the adjoint (or Hermitian transpose) of A. An operator \(B\) is orthogonal if it preserves the inner product, i.e., \[\langle Bv, Bw \rangle = \langle v, w \rangle\] for all vectors \(v, w\) in the Euclidean space. This condition is equivalent to \(B^*B = I\), where \(I\) is the identity operator.

Show that \(e^{A^*} = e^{-A}\)

Recall the definition of the matrix exponential: \[e^A = \sum_{n=0}^{\infty} \frac{A^n}{n!}\] Using the given skew-symmetry of \(A\), we have \[A^* = -A\] Thus, we can calculate \(e^{A^*}\) as follows: \[e^{A^*} = \sum_{n=0}^{\infty} \frac{(A^*)^n}{n!} = \sum_{n=0}^{\infty} \frac{(-A)^n}{n!} = e^{-A}\]

Prove that \((e^A)^*(e^A) = I\)

To show that \(e^A\) is orthogonal, we need to prove that \((e^A)^*(e^A) = I\). Using the result from Step 2, we get \[(e^A)^*(e^A) = e^{-A} e^A\] Since \(A\) and \(-A\) commute, we can use the property of matrix exponentials which states that, whenever \(M\) and \(N\) commute, \(e^{M+N} = e^M e^N\). Thus, in our case, \[e^{-A}e^A = e^{-A+A} = e^0 = I\]

Conclude that \(e^A\) is orthogonal

We have shown that \((e^A)^*(e^A) = I\), which is the condition for an operator to be orthogonal. Thus, if the operator \(A\) in a Euclidean space is skew-symmetric, the operator \(e^A\) is orthogonal.

Key concepts

Skew-Symmetric Operator

In the realm of linear algebra, understanding the nature of operators like skew-symmetric operators is crucial, especially in Euclidean spaces. A skew-symmetric operator is a linear operator that satisfies the condition \( A = -A^* \). Here, \( A^* \) stands for the adjoint or the Hermitian transpose of the operator \( A \). This essentially means that transposing and then taking the complex conjugate of \( A \) results in \(-A\), ensuring that the matrix is the negative of its transpose.

Key points about skew-symmetric operators include:

• All the diagonal elements of a skew-symmetric matrix are zero.
• The non-diagonal elements are such that the transpose changes their signs.

Understanding these properties is fundamental because they lead to interesting results when we explore other mathematical constructs, such as the matrix exponential.

Orthogonal Operator

Orthogonality is a pivotal concept in linear algebra, especially when dealing with transformations in Euclidean spaces. An orthogonal operator is one that preserves the inner product in the space, meaning \( B \) satisfies \( \langle Bv, Bw \rangle = \langle v, w \rangle \) for all vectors \( v \) and \( w \). This is equivalent to the condition \( B^*B = I \), where \( I \) is the identity matrix. In simple terms, an orthogonal operator preserves vector lengths and angles between vectors.

Key features include:

• Orthogonal operators have determinant \( \pm 1 \).
• They are often used in rotations and reflections.

The preservation properties make orthogonal operators invaluable in applications such as computer graphics, where maintaining exact angles and distances is crucial.

Matrix Exponential

The matrix exponential is a fascinating operation that provides solutions to matrix differential equations, among other things. For a matrix \( A \), the exponential \( e^A \) is defined as \( e^A = \sum_{n=0}^{\infty} \frac{A^n}{n!} \). This infinite series representation is crucial for understanding transformations in various spaces.

The matrix exponential has several important properties:

• If \( A \) is skew-symmetric, then \( e^{A^*} = e^{-A} \).
• When matrices \( M \) and \( N \) commute, \( e^{M+N} = e^M e^N \).

The matrix exponential's ability to express transformations in such a concise form makes it a powerful tool for applications ranging from quantum mechanics to computer graphics.