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Due to its simple yet elegant structure, the study of an entry-wise product of matrices, called the Hadamard product, has received extensive attention from researchers and has expanded to various disciplines, including Euclidean Jordan algebras. As an ongoing effort to extend this product to Euclidean Jordan algebras, in this article, we propose a Hadamard product in the setting of Jordan spin algebra,, under the scheme of the Peirce decomposition, and show that it preserves the diagonal structure of the elements in the algebra. It is shown that this new product corresponds to the standard Hadamard product ofsymmetric matrices in the case of. Lastly, we prove that this novel product satisfies an analog of the Schur product theorem as well as the inequalities of Hadamard, Oppenheim, Fiedler, etc.

Hadamard product and related inequalities in the Jordan spin algebra