ObservablesThe Pauli OperatorsOuter ProductsThe Closure RelationRepresentations of Operators Using MatricesOuter Products and Matrix RepresentationsMatrix Representation of Operators in Two-Dimensional SpacesDefinition: The Pauli MatricesHermitian, Unitary, And Normal OperatorsEigenvalues and EigenvectorsSpectral DecompositionThe Trace Of An OperatorImportant Properties of the TraceThe Expectation Value of an OperatorProjection OperatorsPOSITIVE OPERATORSCommutator AlgebraThe Heisenberg Uncertainty PrinciplePolar Decomposition and Singular ValuesThe Postulates of Quantum MechanicsExercises