Is a Matrix of Only Diagonal Ones Always Equal to its Transpose?

[mikedamike]

Very quick question.

Im 99% sure that the transpose of a matrix of only diagonal ones is always equal to its original state.

Ie
|1000| |1000| T
|0100| |0100|
|0010| = |0010|
|0001| |0001|


So my question is am i correct ?

Thanks in advance.

 

[tiny-tim]

hi mikedamike!

Very quick question.

Im 99% sure that the transpose of a matrix of only diagonal ones is always equal to its original state.

Ie

              |1000|       |1000| T
              |0100|       |0100|
              |0010| =    |0010|
              |0001|       |0001|

So my question is am i correct ?

Thanks in advance.

yes, and it works for any diagonal matrix …

(a)^T_ii = (a)_ii ​

 

[HallsofIvy]

The definition of "transpose" is that if the elements of A are "$a_{ij}$" then the elements of A^T are $a_{ji}$. So if all non-diagonal elements are 0, we have $a_{ij}= a_{ji}= 0$. And, of course, on the diagonal i= j so we still have $a_{ij}= a_{ji}$.