An inequality about linear transformations

Question

Is it true that : $$\inf_{x^2+y^2=1}\left(\left|\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}a'&b'\\c'&d'\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\right|\right)\geq\inf_{x^2+y^2=1}\left|\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\right|\cdot\inf_{x^2+y^2=1}\left|\begin{pmatrix}a'&b'\\c'&d'\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\right|$$ where all the variables are reals and $|.|$ denotes the euclidean norm. Can someone give a counterexample or confirm that this is true?

Answer 1

Yes, it is true. Let $$M_1=\inf_{x^2+y^2=1}\left\lVert\begin{bmatrix}a&b\\c&d\end{bmatrix}.\begin{bmatrix}x\\y\end{bmatrix}\right\rVert\text{ and let }M_2=\inf_{x^2+y^2=1}\left\lVert\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}.\begin{bmatrix}x\\y\end{bmatrix}\right\rVert.$$

Let$$\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}.\begin{bmatrix}x\\y\end{bmatrix}.$$

Then\begin{align}\left\lVert\begin{bmatrix}a&b\\c&d\end{bmatrix}.\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}.\begin{bmatrix}x\\y\end{bmatrix}\right\rVert&=\left\lVert\begin{bmatrix}a&b\\c&d\end{bmatrix}.\begin{bmatrix}x'\\y'\end{bmatrix}\right\rVert\\&\geqslant M_1.\left\lVert\begin{bmatrix}x'\\y'\end{bmatrix}\right\rVert\\&=M_1.\left\lVert\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}.\begin{bmatrix}x\\y\end{bmatrix}\right\rVert\\&\geqslant M_1.M_2.\left\lVert\begin{bmatrix}x\\y\end{bmatrix}\right\rVert\end{align} And now taking the infimum over the unit circle we have what we wanted.