- #1

- 274

- 1

## Homework Statement

I have a linear transformation T defined by

$$

T(v_{1})=v_{1}+iv_{2}\\

T(v_{2})=-iv_{1}+v_{2}\\

$$

and I want to find a triangular matrix B of T and an invertible matrix S such that SB=AS where A is the matrix of T with respect to the basis ##\{v_{1},v_{2}\}##.

## The Attempt at a Solution

The matrix A is

$$

\begin{pmatrix}

1 & -i \\

i & 1 \\

\end{pmatrix}.

$$

From the proof of the triangular form theorem, I know that I need to find a new basis ##\{w_{1},w_{2}\}## such that ##Tw_{1}=0## and ##(T-2)w_{2}=0## since the minimal polynomial is ##m(T)=T(T-2)=0## but I must be messing something up because there is no way I can create a triangular matrix B from ##w_{1}## and ##w_{2}##. If I find a linear combination of ##v_{1},v_{2}## equal to ##w_{1}## such that ##Tw_{1}=0## then the first column of my matrix B would always be 0.