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Math Help - Last Linear Algebra problem

  1. #1
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    Last Linear Algebra problem

    Defenition: Let A and B be nxn matrices. Then A is said to be similar to B, denoted A is congruent to B, if there exists an invertible nxn matrix P such that B=(P^-1)AP.

    a. Let A=I_n. Prove that, if A is congruent to B, then B=I_n

    b. Let A be a nxn idempotent matrix. Prove that if A is congruent to B, then B is idempotent.

    c. Is the analogue of part b above the nilpotent matrices true? Why or Why not?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by natewalker205 View Post
    Defenition: Let A and B be nxn matrices. Then A is said to be similar to B, denoted A is congruent to B, if there exists an invertible nxn matrix P such that B=(P^-1)AP.

    a. Let A=I_n. Prove that, if A is congruent to B, then B=I_n

    b. Let A be a nxn idempotent matrix. Prove that if A is congruent to B, then B is idempotent.

    c. Is the analogue of part b above the nilpotent matrices true? Why or Why not?

    In all of these cases you just need to observe that as A and B are congruent there exists an invertible matrix P such that:

    <br />
B=P^{-1}AP<br />

    Now use what you are told in a, b and c and the answers follow at once

    RonL
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