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We present an orthogonal basis of gauge invariant operators constructed fromsome complex matrices for the free matrix field, where operators are expressedwith the help of Brauer algebra. This is a generalisation of our previous workfor a signle complex matrix. We also discuss the matrix quantum mechanicsrelevant to N=4 SYM on S^{3} times R. A commuting set of conserved operatorswhose eigenstates are given by the orthogonal basis is shown by using enhancedsymmetries at zero coupling.

Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra