I have a square matrix A over \mathbb{C} then is

Gabriela Duarte

Gabriela Duarte

Answered question

2022-01-31

I have a square matrix A over C then is its rank invariant under a congruence transformation APtAP ? What's the easiest way to see this?

Answer & Explanation

tsjutten20

tsjutten20

Beginner2022-02-01Added 13 answers

Step 1 Suppose M is invertible, then the vectors v1,...,vk are linearly independent iff the vectors Mv1,...,Mvk are. The same applies to M1, of course. It is easy to see that RC=R(CM) for any matrix C. It is also easy to see that R(MC)=M(RC). In particular, the above shows that dimR(MC)=dimRC In the above case this gives R(PTAP)=PTR(AP)=PTRA, hence dimR(PTAP)=dimRA..
search633504

search633504

Beginner2022-02-02Added 16 answers

This is true if P is invertible. Noting that rank(AB)min{rank(A),rank(B)} we have rank(AB)=rank(BA)=rank(A) whenever B is invertible. Thus, rank(PTAP)=rank(PTA)=rank(A)

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