# Choosing the pivot for the rotation matrix in similarity transformation

I have arrived at an equation in the similarity transformation -

$$M_r$$ = $$T. M_{r-1}. T^t$$ ,where $$T$$ is the rotation matrix and $$M_r$$ ,$$M_{r-1}$$ are similar matrices. My aim is to find the rotation matrix $$T$$ with a pivot element $$(a,b)$$ such that all elements $$m_{ij} \in M$$ will be as follows :

$$m_{ij} \neq 0$$ , if $$|i-j| = 1$$

$$m_{ij} \neq 0$$ , if $$i=j$$ , i.e, it lies on the principal diagonal

I found the similarity transformation of a matrix in an application for Microwave filter design by Richard J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions" where they used similarity transformation using rotation matrix for annihilation of elements. But the concept of setting the pivot is not very clear to me.

Can someone explain how the pivot is set to eliminate a particular element of $$M$$, for instance?

• Is $M_0$ symmetric? – Federico Poloni Apr 29 '20 at 12:52
• Anyway, at a quick glance this looks like a variant of tridiagonal reduction (see e.g. [Golub, Van Loan, IV edition, Sec. 8.3.1]), only that Givens transformations are used instead of Householder ones. – Federico Poloni Apr 29 '20 at 13:08
• Yes $M_0$ is symmetric. – Divyayan Dey Apr 29 '20 at 14:27
• And there must be something wrong in your formulas: the first requirement tells you $m_{11}=0$, the third one $m_{11}\neq 0$. – Federico Poloni Apr 29 '20 at 14:36
• Sorry about that. Edited. – Divyayan Dey Apr 29 '20 at 15:25