Let A ∈ M n ( R ) such that A 2 = − I n and AB=BA for some B ∈ M n ( R ). Prove that det ( B ) ≥ 0.All the information I could extract from the relation A 2 = − I n are as follows:(a) A is not diagonalizable.(b) det(A)=1.(c) n must be even.How to conclude that det(B) is nonnegative using these 3 informations alongwith AB=BA ?

Question

Let   A  ∈      M    n    (      R    ) such that       A    2    =  −      I    n   and AB=BA for some   B  ∈      M    n    (      R    ). Prove that   det  (  B  )  ≥  0.All the information I could extract from the relation       A    2    =  −      I    n   are as follows:(a) A is not diagonalizable.(b) det(A)=1.(c) n must be even.How to conclude that det(B) is nonnegative using these 3 informations alongwith AB=BA ?

doraemonjrlf · Accepted Answer

Proof Outline: Using the fact that       A    2    =  −      I    n  , conclude that n must be even and that there exists some invertible matrix   P  ∈      M    n    (      R    ) such that      P          −      1        A  P  =  J  :=      (                            0                          −                      I            k                                                            I            k                                    0                      )    ,where   k  =  n      /    2. With that, we can conclude that det(A)=1.Now without loss of generality, we can assume that A=J (note that A commutes with B iff       P          −      1        A  P commutes with       P          −      1        B  P). Partition B into four   k  ×  k blocks:  B  =      (                                        B                          11                                                            B                          12                                                                        B                          21                                                            B                          22                                            )    .From the fact that   A  B  =  B  A (that is,   J  B  =  B  J), conclude that we have       B          11        =      B          22       and       B          12        =  −      B          21      . That is, we have  B  =      (                            F                          −          G                                      G                          F                      )  for some matrices   F  ,  G  ∈      M    k    (      R    ). Now, find a matrix   Q  ∈      M    n    (      C    ) such that      Q          −      1        B  Q  =      (                            F          +          i          G                          0                                      0                          F          −          i          G                      )    .Conclude that                    det        (        B        )                            =        det        (        F        +        i        G        )        det        (        F        −        i        G        )        =        det        (        F        +        i        G        )        det        (                              F            +            i            G                    ¯                )                                          =        det        (        F        +        i        G        )                              det            (            F            +            i            G            )                    ¯                =                  |                det        (        F        +        i        G        )                              |                    2                ≥        0.

Let A in M_n(RR) such that A^2=-I_n and AB=BA for some B in M_n(RR). Prove that det(B) >=0.

Answered question

Answer & Explanation

New Questions in Precalculus