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Prove that diagonal matrices are symmetric matrices

  1. Oct 9, 2016 #1
    1. The problem statement, all variables and given/known data
    Same as title.

    2. Relevant equations


    3. 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?
     
  2. jcsd
  3. Oct 9, 2016 #2

    fresh_42

    Staff: Mentor

    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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