Question

If the linear transformation \( X = BY \) transforms \( X^\top AX \) to \( Y^\top PY \), then \( P = \):

• A. \( BAB \), if \( B \) is an orthogonal matrix
• B. \( BAB \), if \( B \) is a symmetric matrix
• C. \( B^{-1}AB \), if \( B \) is a symmetric matrix
• D. \( A^{-1}BA \), if \( A \) is a non-singular matrix

Hint:

When working with quadratic forms and linear transformations, always remember: - Substitute the transformation directly, - Use transpose rules carefully, - Apply symmetry or orthogonality conditions to simplify.

Answer 1

The Correct Option is B

Step 1: Use the given transformation \( X = BY \).
We are given that: \[ X^\top AX = Y^\top PY \] Substitute \( X = BY \): \[ (BY)^\top A (BY) = Y^\top P Y \] Since \( (BY)^\top = Y^\top B^\top \), we get: \[ Y^\top B^\top A B Y = Y^\top P Y \]
Step 2: Compare both sides.
We now compare the quadratic forms: \[ Y^\top B^\top A B Y = Y^\top P Y \Rightarrow P = B^\top A B \]
Step 3: Use symmetry condition.
If \( B \) is symmetric, then \( B^\top = B \). Thus, \[ P = B A B \]