Define a relation R over a class of n [] n real matrices A and B as "ARB iff there exists …

Question

Define a relation R over a class of n $$\times$$ n real matrices A and B as

"ARB iff there exists a non-singular matrix P such that PAP-1 = B".

Then which of the following is true?

Accepted Answer

For reflexive relation,

$\forall(A, A) \in R$ for matrix $P$.

$\Rightarrow A=P A P^{-1}$ is true for $P=1$

So, $R$ is reflexive relation.

For symmetric relation,

Let $(A, B) \in R$ for matrix $P$.

$$
\Rightarrow \quad A=P B P^{-1}
$$

After pre-multiply by $P^{-1}$ and post-multiply by $P$, we get

$P^{-1} A P=B$

So, $(B, A) \in R$ for matrix $P^{-1}$.

So, $R$ is a symmetric relation.

For transitive relation,

Let $A R B$ and $B R C$

So, $A=P B P^{-1}$ and $B=P C P^{-1}$

Now, $A=P\left(P C P^{-1}\right) P^{-1}$

$\Rightarrow A=(P)^2 C\left(P^{-1}\right)^2 \Rightarrow A=(P)^2 \cdot C \cdot\left(P^2\right)^{-1}$

$\therefore(A, C) \in R$ for matrix $P^2$.

$\therefore R$ is transitive relation.

Hence, $R$ is an equivalence relation.

Solutions · 1