| course |
course_year |
question_number |
tags |
title |
year |
Quadratic Mathematics |
IB |
63 |
IB |
2002 |
Quadratic Mathematics |
|
1.II.17F |
2002 |
Let $A$ and $B$ be $n \times n$ real symmetric matrices, such that the quadratic form $\mathbf{x}^{T} A \mathbf{x}$ is positive definite. Show that it is possible to find an invertible matrix $P$ such that $P^{T} A P=I$ and $P^{T} B P$ is diagonal. Show also that the diagonal entries of the matrix $P^{T} B P$ may be calculated directly from $A$ and $B$, without finding the matrix $P$. If
$$A=\left(\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 2
\end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc}
4 & 0 & 0 \\
0 & -2 & 0 \\
0 & 0 & 0
\end{array}\right)$$
find the diagonal entries of $P^{T} B P$.