Prove that diagonal matrices are symmetric matrices

  • #1
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Homework Statement


Same as title.

Homework Equations




The Attempt at a Solution


A defining property of a diagonal matrix is that ##A_{ij} = A_{ji} ~~\forall i,j \le n##. This means that ##((A)^{t})_{ji} = A_{ji}##. Therefore, we know that ##A^t = A##. This shows that a diagonal matrix is symmetric.

Is this an okay proof? Am I making too big of a leap in logic to start with ##A_{ij} = A_{ji} ~~\forall i,j \le n##? Or do I need to first prove that that statement is true for diagonal matrices?
 

Answers and Replies

  • #2
15,118
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I would say, in the case of a diagonal matrix, there is nothing to prove, since all ##A_{ij} = 0 = A_{ji}## for ##i \neq j## and of course is ##A_{ii}=A_{ii}## for the rest.
 

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