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For a linear mapping $L$ in the plane transforming $[1 ,0]$ to $[a ,c]$ and $[0 ,1]$ to $[b, d]$ a non-zero vector $v$ such that $v$ and $Lv$ are orthogonal will be named an ortho-eigen vector for $L$. May be there is another name for such vector: my name goes in parallel with an eigen vector, which you get by replacing "orthogonal" by parallel. A task: Find conditions on $a,b,c,d$ to determine if $L$ has ortho-eigen vectors.

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Assuming the usual euclidean product:

So we can put $\,L\,$ in matrix form wrt the standard basis of $\,\Bbb R^2\,$ :

$$[L] =\begin{pmatrix}a&b\\c&d\end{pmatrix}$$

and then we have an orthoeigenvector iff

$$0=\left\langle\;\binom xy\;,\;\binom{ax+by}{cx+dy}\;\right\rangle=ax^2+(b+c)xy+dy^2$$

Let us look at the above right side of the equation as a quadratic in $\,x\,$ :

$$ax^2+(b+c)y\,x+dy^2=0\;\;\text{has a real solution iff}$$

$$\Delta=(b+c)^2y^2-4ady^2\ge 0\iff(b+c)^2y^2\ge4ady^2$$

Now check cases: if $\,y=0\,$ we have an obvious solution whenever $\,ax=0\,$ , otherwise

$$y\neq 0\implies(b+c)^2\ge4ad\implies (b+c)^2-4bc\ge 4(ad-bc)\iff$$

$$\text{tr.}^2[L]-4bc\ge4\det[L]\;\ldots\ldots$$

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