# How to determine two matrices are conjugate.

Which of the following statements are true?

• The matrices $A=\left[ {\begin{array}{cc} 1 & 1 \\ 0 & 1\\ \end{array} } \right]$ and $B=\left[ {\begin{array}{cc} 1 & 0 \\ 1 & 1\\ \end{array} } \right]$ are conjugate in $GL_2(\mathbb{R})$
• The matrices $A=\left[ {\begin{array}{cc} 1 & 1 \\ 0 & 1\\ \end{array} } \right]$ and $B=\left[ {\begin{array}{cc} 1 & 0 \\ 1 & 1\\ \end{array} } \right]$ are conjugate in $SL_2(\mathbb{R})$
• The matrices $C=\left[ {\begin{array}{cc} 1 & 0 \\ 0 & 2\\ \end{array} } \right]$ and $D=\left[ {\begin{array}{cc} 1 & 3 \\ 0 & 2\\ \end{array} } \right]$ are conjugate in $GL_2(\mathbb{R})$

I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.

## Hint

The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^{-1}BP$$ then they are conjugate. In $GL(2,\mathbb{R})$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,\mathbb{R})$ have the same Jordan normal form

Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable

Probabily this forum Why aren't this two matrices conjugate in $SL(2,\mathbb{R})$ can help for the second question.

• What is $D$ here? – Babai Jul 23 '18 at 9:15
• Sorry, I'm indicating the matrices as they are on your worksheet $$A=\left[\begin{matrix} 1&1\\0&1\end{matrix}\right]$$ and $$D=\left[\begin{matrix} 1&3\\0&2\end{matrix}\right]$$ – Davide Morgante Jul 23 '18 at 9:17