Simple Matrix Calculation

Question

I am new to Math, and my knowledge is not good. Can anyone help me with this problem? Show that $(M^*)^* = M$, $(M^T)^T = M$, and $(M^t)^t = M$. I think I got some idea on the last 2, but the I am stuck at the first one.

@Sopheak You probably mean conjugate transpose. This should be useful: $z = a+bi$ is a complex number, then the complex conjugate of the complex conjugate of $z$ is $z$ itself. — Gyu Eun Lee, Mar 17 '14 at 08:03
Thank you. But I think I am dealing with Complex Conjugate of Matrix. — TBBT, Mar 17 '14 at 08:06
@AnonSubmitter85: I don't know how to type my idea in comment section. Sorry. — TBBT, Mar 17 '14 at 08:12
Let the entry in row $i$, column $j$ be $a_{ij}$. Then work out the entry in row $i$, column $j$ of $(M^)^$ and the others. — Gerry Myerson, Mar 17 '14 at 08:15
If $A^\ast$ is just $A$ with each entry conjugated, then won't $(A^\ast)^\ast$ just be $A$ with each entry conjugated twice? — AnonSubmitter85, Mar 17 '14 at 08:16

score 0 · Answer 1 · answered Mar 17 '14 at 11:42

As an example of the kind of thinking you need: Let $A^T$ be the transpose of the matrix $A$. That is, you get $A^T$ from $A$ by switching rows and columns: Take the first row of $A$, write it down as a column (the first one) . Then take the second row of $A$, and write it down as a column, to the right of the first column you wrote down. Continue that way. When finnished in that way with all the rows, what you have written down is the transpose of $A$.

Now repeat the same procedure: Take the first row of $A^T$, write it down as a column. Continue with second row, ...

When terminated, what you have should be identical to the matrix $A$ with which you started. You have shown that $(A^T)^T = A$.

Thank you for the answer. That helps me alots. – TBBT Mar 18 '14 at 04:29 — TBBT, Mar 18 '14 at 04:29

Simple Matrix Calculation

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