Let V be the vector space of all n x n matrices over the field of complex number
ID: 2938606 • Letter: L
Question
Let V be the vector space of all n x n matrices over the field of complex numbers, with the inner product . ( A | B ) = tr( AB* ). . . Find the orthogonal complement of the subspace of diagonalmatrices. . . Note the orthogonal complement of set S in V is the setof vectors in V which are orthogonal to every vector in S . Thanks for looking. Promise to rate. with the inner product . ( A | B ) = tr( AB* ). . . Find the orthogonal complement of the subspace of diagonalmatrices. . . Note the orthogonal complement of set S in V is the setof vectors in V which are orthogonal to every vector in S . Thanks for looking. Promise to rate.Explanation / Answer
If matrix A has values a1, a2, ..., an on the diagonal, then leftmultiplication by A on a matrix B* multiplies the 1st row of B* bya1, the second by a2, etc. Thus the trace of AB* is a1b1 +a2b2 + ... + anbn, where b1, b2, ..., bn are the diagonal entriesof B*. If B is in the orthogonal complement to all thediagonal matrix, then this trace must be 0 for every possible valueof a1, a2, ..., an. Letting a1 = 1, and all the rest of theai = 0 shows that b1 must be 0. Letting a2 = 1 and all therest of the ai = 0 shows b2 = 0, etc. Thus, B* must have all0's on the diagonal. Hence, B must be the same as well. So, the orthogonal complement to the diagonal matrices is the setof matrices with all 0's on the diagonal.
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