Matrix-Trace and Conjugate Transpose (Multiple Choice)

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Only 2 is false. You can easily decide on all four by using $$ \text{Tr}(A^*A)=\sum_{j=1}^n\sum_{k=1}^n|A_{kj}|^2. $$

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usermath
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usermath

Updated on August 01, 2022

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  • usermath
    usermath over 1 year

    I was trying to solve the following problem from a competitive exam paper.

    Let $A=( a_{ij})$ be a nXn complex matrix and let $A^*$ denote the conjugate transpose of $A$. Then which of the following statements are necessarily true? (One or more options may be correct)

    1. $A^{-1}$ exists $\Rightarrow tr(A^*A)\neq 0 $
    2. $ tr(A^*A)\neq 0 \Rightarrow A^{-1} $ exists.
    3. $|tr(A^*A)|<n^2\Rightarrow | a_{ij}|<1 $ for some $i,j$
    4. $ tr(A^*A)= 0 \Rightarrow A = 0$

    I am completely stuck.

    Please help me. Thnx.

  • usermath
    usermath over 9 years
    Thank you very much for your help. I have a little doubt. Can you please clarify how option 3 is coming out to be true from your given formula?
  • Martin Argerami
    Martin Argerami over 9 years
    If $|A_{kj}|\geq1$ for all $k,j$, then adding $n^2$ of those numbers will be at least $n^2$.
  • usermath
    usermath over 9 years
    Thank you again.It is very much clear now.