Matrix-Trace and Conjugate Transpose (Multiple Choice)
1,952
Only 2 is false. You can easily decide on all four by using $$ \text{Tr}(A^*A)=\sum_{j=1}^n\sum_{k=1}^n|A_{kj}|^2. $$
Author by
usermath
Updated on August 01, 2022Comments
-
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)
- $A^{-1}$ exists $\Rightarrow tr(A^*A)\neq 0 $
- $ tr(A^*A)\neq 0 \Rightarrow A^{-1} $ exists.
- $|tr(A^*A)|<n^2\Rightarrow | a_{ij}|<1 $ for some $i,j$
- $ tr(A^*A)= 0 \Rightarrow A = 0$
I am completely stuck.
Please help me. Thnx.
-
usermath over 9 yearsThank 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 over 9 yearsIf $|A_{kj}|\geq1$ for all $k,j$, then adding $n^2$ of those numbers will be at least $n^2$.
-
usermath over 9 yearsThank you again.It is very much clear now.