A diagonal matrix and a symmetric matrix commute if their product is itself a symmetric matrix and vice versa (see the answers to this question). Are there other conditions that make their product symmetric?
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If the entries of the diagonal matrix are distinct, then it commutes with a symmetric matrix if and only if that matrix is diagonal.
More generally, if the diagonal matrix has the form $$ D = \pmatrix{\lambda_1 I_{k_1}\\&\lambda_2 I_{k_2} \\ && \ddots \\ &&& \lambda_m I_{k_m}} $$ where $I_k$ denotes the $k \times k$ identity matrix, then a (symmetric) matrix $A$ will commute with $D$ if and only if $A$ is conformally block diagonal, which is to say that $$ A = \pmatrix{A_1 \\ & A_2 \\ & & \ddots \\ &&& A_m} $$ Where for each $j$, $A_j$ is a (symmetric) matrix of size $k_j \times k_j$.
Ben Grossmann
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When A is symmetric, $A_j$ is also symmetric, right? – Abhiram V P May 24 '19 at 05:08
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The only diagonal matrices such that their product with every symmetric matrix is again symmetric are the multiples of the identity matrix.
José Carlos Santos
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I am not interested in the product with every symmetric matrix. Only the special cases when this occurs. – Tarek Jul 19 '17 at 13:28