Question

# If q is a quadratic form on $\mathbb{R}^{n}$ with symmetric matrix A and if $L(\vec{x})=R \vec{x}$ is a linear transformation from $\mathbb{R}^{m}$ to $\mathbb{R}^{n},$ show that the composite function $p(\vec{x})=q(L(\vec{x}))$ is a quadratic form on $\mathbb{R}^{m}.$ Express the symmetric matrix of p in terms of R and A.

Solution

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Using the definitions of the functions $p, L$ and the quadratic form $q$ defined by the symmetric matrix $A,$ we have

\begin{align*} p(\vec{x})&=q(L(\vec{x})) \\ &=q(R\vec{x})\\ &=(R \vec{x})^T (A) (R\vec{x}) \\ &=(\vec{x}^TR^T)(A)(R \vec{x})\\ &=\vec{x}^T (R^TAR) \vec{x}\\ &=\vec{x}^T B \vec{x} \end{align*}

where $B=R^TAR$ is an $m \times m$ symmetric matrix as

$B^T=(R^TAR)^T=R^TA^T(R^T)^T=R^TAR$ (since $A$ is symmetric).

Thus, $p(\vec{x})=\vec{x}^T B \vec{x}$ is a quadratic form in $m$ variables defined by the symmetric matrix $B=R^TAR.$

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