= Similarity and the coordinate metric
{title2=$G=S^TS,\quad B^TG=GB$}
For real <matrices>, <orthogonal similarity> is the special case $B=Q^{-1}AQ=Q^TAQ$ with $Q$ an <orthogonal matrix>. Then a <symmetric matrix> remains symmetric, because $B^T=Q^TA^TQ=B$. The expression $QAQ^T$ describes the reverse conversion and has the same property. An <orthonormal basis> retains the ordinary <dot product> formula for its <coordinate vectors>.
An arbitrary <change of basis> need not preserve entrywise symmetry. In the opening example, $A$ is a <symmetric matrix> but $B=\begin{pmatrix}2&-3\\0&5\end{pmatrix}$ is not. The geometric <inner product> has not disappeared: in the new <coordinates> it is represented by the <Gram matrix>
$$
G=S^TS=\begin{pmatrix}1&1\\1&2\end{pmatrix},\qquad
(Sy)^T(Sz)=y^TGz.
$$
For $p$ with new <coordinate vector> $y=(1,1)^T$, the squared <Euclidean norm> is $y^TGy=5$, agreeing with $x^Tx=2^2+1^2$. The naive coordinate sum $y^Ty=2$ would use the wrong <inner product>.
If $A^T=A$, direct multiplication gives
$$
B^TG=GB=S^TAS.
$$
This is <self-adjointness> expressed in the new <inner product>, rather than entrywise symmetry relative to the <identity matrix>. For the example, both products equal $\begin{pmatrix}2&2\\2&7\end{pmatrix}$. Thus arbitrary <matrix similarity> preserves the <linear operator>, while its geometric interpretation requires transforming the <Gram matrix> as well. <Matrix congruence> supplies the corresponding rule for a <bilinear form> and should be distinguished from <matrix similarity>.
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