Question

Let \( \mathbf{v} \in \mathbb{R}^k \) with \( \mathbf{v}^T \mathbf{v} \neq 0 \). Let \[ P = I - 2 \frac{ \mathbf{v} \mathbf{v}^T }{ \mathbf{v}^T \mathbf{v} }, \] where \( I \) is the \( k \times k \) identity matrix. Then which of the following statements is (are) TRUE?

• A. \( P^{-1} = I - P \)
• B. -1 and 1 are eigenvalues of \( P \)
• C. \( P^{-1} = P \)
• D. \( (I + P)\mathbf{v} = \mathbf{v} \)

Hint:

Reflection matrices have the property that \( P^{-1} = P \) and they have eigenvalues of 1 and -1. The matrix \( I + P \) does not necessarily leave the vector \( \mathbf{v} \) unchanged.

Answer 1

The Correct Option is B, C

Step 1: Understanding the matrix \( P \).
The matrix \( P \) is a reflection matrix, often called a Householder transformation, which reflects a vector across the hyperplane orthogonal to \( \mathbf{v} \). The key property of a reflection matrix is that it is idempotent and symmetric, i.e., \( P = P^{-1} \).
Step 2: Analyzing the options.
- (A) \( P^{-1} = I - P \): This is incorrect because \( P = P^{-1} \) by the property of a reflection matrix.
- (B) -1 and 1 are eigenvalues of \( P \): This is correct because for a reflection matrix, one eigenvalue is 1 (along the direction of \( \mathbf{v} \)) and the other is -1 (in the orthogonal direction).
- (C) \( P^{-1} = P \): This is correct because the reflection matrix is an involution, meaning \( P = P^{-1} \).
- (D) \( (I + P)\mathbf{v} = \mathbf{v} \): This is incorrect. The vector \( \mathbf{v} \) is reflected by \( P \), but \( I + P \) does not leave \( \mathbf{v} \) unchanged. Instead, it reflects \( \mathbf{v} \) and adds \( \mathbf{v} \).
Step 3: Conclusion.
Thus, the correct answers are (B) and (C).