Prove symmetric matrix properties
By John Campbell •
(a)Prove sum and difference of two symmetric matrices are also symmetric.
(b)Prove $A^n$ is a symmetric matrix if $A$ is a symmetric matrix
For (a), let matrices $A$ and $B$ be two symmetric matrices such that $A=A^T$ and $B=B^T$
$$A+B=A^T+B^T=(A+B)^T$$$$A+B=(A+B)^T$$
Therefore, sum of two symmetric matrices are also symmetric
Similarly, $$A-B=A^T-B^T=(A-B)^T$$$$A-B=(A-B)^T$$
Therefore, the difference of two symmetric matrices are also symmetric
For(b), if $A$ is a symmetric matrix, $$A=A^T$$ $$A^n=(A^T)^n$$
Note that by properties of the transpose of the matrix, $$(AB)^T=B^TA^T$$ thus $$A^n=AA...A$$ $$(A^n)^T=A^TA^T...A^T$$ $$(A^n)^T=(A^T)^n$$
Therefore, $$A=A^T$$ $$A^n=(A^T)^n$$ $$A^n=(A^n)^T$$
Do I make any mistake(s)?
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