How to determine two matrices are conjugate.

Question

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.

Answer 1

Quiver has already told you (and I am sure it is in you textbook) that two matrices, A and B, are conjugate if and only if there exist an invertible matrix, P, such that $A= P^{-1}BP$. That is equivalent to $PA= BP$.

In the first problem, $A= \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}$ and $B= \begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$. We can show they are conjugate by finding an appropriate P!

Since A and B are 2 by 2 P must be also and we can write it $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ so we must have $\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$\iff$ $\begin{bmatrix}a & a+ b\\ c & c+ d\end{bmatrix}= \begin{bmatrix}a & b \\ a+ c & b+ d\end{bmatrix}$.

We must have a= a, a+ b= b, c= a+ c, and c+ d= c+ d. Both a+ b= b and c= a+ c reduce to a= 0 while c+ c= c+ d is always true. Any matrix of the form $\begin{bmatrix}0 & b \\ c & d\end{bmatrix}$ will do.

Answer 2

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 matrix $D$ is diagonalisable.

Probabily this forum can help for the second question.