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The Hadamard Product - UPS

TheHadamardProductElizabeth MillionApril 12, 20071 Introduction and Basic ResultsAs inexperienced mathematicians we may have once thought that the natural definition for matrixmultiplication would be entrywise multiplication, much in the same way that a young child mightsay, I writed my name. The mistake is understandable, but it still makes us cringe. Unlike poorgrammar, however, entrywise matrix multiplication has reason to be studied; it has nice propertiesin matrix analysis and has applications in both statistics (301 [4], 140 [5]) and physics (93, 149 [5]).Here we will only expore the properties of the Hadamard Product in matrix nmatrices with entries inC. TheHadamard productof Aand B is defined by[A B]ij= [A]ij[B]ijfor all1 i m,1 j we can see, the Hadamard Product is simply entrywise multiplication. Because of this, theHadamard Product inherits the same benefits (and restrictions) of multiplication inC.

The Hadamard identity matrix and the Hadamard inverse are both more limiting than helpful, so we will not explore their use further. One last fun fact: the set of m ×n matrices with nonzero entries form an abelian (commutative) group under the Hadamard product (Prove this!). Theorem 1.5. The Hadamard Product is Linear.

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