Prove that every diagonal matrix is a symmetric matrixSoluti

Prove that every diagonal matrix is a symmetric matrix

Solution

Let us consider A a square nxn matrix which is also diagonal.

i.e A = {a_ij} where a_ij =0 if i not equal to j

= some non zero value is i =j

Transpose of A = A^T= {a_ji}

But we have a_ji =0 if i not equals j and a_ji non zero if i =j

In other words A^T will have all elements other than diagonal as zero elements.

Also a_ji =a_ij , if i =j since all these types are diagonal elements.

So A^T = A

Or A is symmetric if A is diagonal.