Orthogonal Matrix Properties Analysis

We are given an $n \times n$ orthogonal matrix $A$ with real entries. By definition, an orthogonal matrix satisfies $A^T A = I$. This property leads to several key characteristics:

• Norm Preservation: For any vector $\mathbf{x}$, the Euclidean norm is preserved: $\|A\mathbf{x}\| = \|\mathbf{x}\|$. This is shown as follows: $ \|A\mathbf{x}\|^2 = (A\mathbf{x})^T (A\mathbf{x}) = \mathbf{x}^T A^T A \mathbf{x} = \mathbf{x}^T I \mathbf{x} = \mathbf{x}^T \mathbf{x} = \|\mathbf{x}\|^2 $
• Inner Product Preservation: The dot product (inner product) between vectors is also preserved: $(A\mathbf{x})^T(A\mathbf{y}) = \mathbf{x}^T\mathbf{y}$. This follows from: $ (A\mathbf{x})^T(A\mathbf{y}) = \mathbf{x}^T A^T A \mathbf{y} = \mathbf{x}^T I \mathbf{y} = \mathbf{x}^T\mathbf{y} $
• Determinant Property: Taking the determinant of $A^T A = I$, we get $\det(A^T A) = \det(I)$. This implies $\det(A^T)\det(A) = 1$. Since $\det(A^T) = \det(A)$, we have $(\det(A))^2 = 1$, which means $\det(A) = +1$ or $\det(A) = -1$.
• Eigenvalue Property: If $\lambda$ is an eigenvalue of $A$ with eigenvector $\mathbf{v}$, then $A\mathbf{v} = \lambda \mathbf{v}$. Using the norm preservation property: $ \|A\mathbf{v}\| = \|\mathbf{v}\| $ $ \|\lambda \mathbf{v}\| = \|\mathbf{v}\| $ $ |\lambda| \|\mathbf{v}\| = \|\mathbf{v}\| $ Since $\|\mathbf{v}\| \neq 0$, we must have $|\lambda| = 1$. Thus, all eigenvalues lie on the unit circle in the complex plane.

Evaluating the Options

1. Statement 1: The value of the determinant of $A$ is either +1 or -1.

   This aligns with the determinant property derived above. Correct.

2. Statement 2: The eigenvalues of $A$ have modulus 1.

   This aligns with the eigenvalue property derived above. Correct.

3. Statement 3: $\|Ax\| = \|x\|$, for all $\mathbf{x} \in \mathbb{R}^n$, and $(A\mathbf{x})^T(A\mathbf{y}) \neq \mathbf{x}^T\mathbf{y}$, for all distinct $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$.

   While $\|Ax\| = \|x\|$ is true, the condition $(A\mathbf{x})^T(A\mathbf{y}) \neq \mathbf{x}^T\mathbf{y}$ is false. Orthogonal matrices preserve the inner product, meaning $(A\mathbf{x})^T(A\mathbf{y}) = \mathbf{x}^T\mathbf{y}$ for all $\mathbf{x}, \mathbf{y}$. Incorrect.

4. Statement 4: $\|Ax\| = \|x\|$, for all $\mathbf{x} \in \mathbb{R}^n$, where $\|\mathbf{x}\|$ denotes the Euclidean norm of $\mathbf{x}$, and $(A\mathbf{x})^T(A\mathbf{y}) = \mathbf{x}^T\mathbf{y}$, for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$.

   Both parts of this statement correctly describe the properties of orthogonal matrices (norm preservation and inner product preservation). Correct.

Therefore, statements 1, 2, and 4 are correct.