Algebraical methods for study of correlational connections in incomplete balanced block design. II. Relationship algebras and characterization of covariance matrices in the case of symmetrical blocks

For incomplete balanced blocks of experiment design the relation of the commutator algebra and the relationship algebra of A. T. James is studied. For symmetrical flowcharts a six-dimensional relationship algebra and a sevendimensional extended relationship algebra are introduced. Covariance matrices belonging to these algebras are characterized. Necessary and sufficient conditions to be satisfied by expansion coefficients of nonegative and positive definite matrices in specially selected bases of these algebras are obtained.