There are four normed division algebras over real numbers, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions. Eigenvalue problem of octonionic Hermitian matrices is one of the interesting studies where we can see this difficulty of extending the basic properties from complex Hermitian matrices to octonionic Hermitian matrices. This includes the notion of orthogonality and decomposition of a Hermitian matrix using its eigenvalues and eigenvectors.Liping Huang and Wasin So derived explicit formulas for computing the roots of quaternionic quadratic equations. We extend their work to octonionic case and solve octonionic left quadratic equations. We represent left spectrum of two by two octonionic Hermitian matrix using the solutions to corresponding octonionic left quadratic equation and identify the family of matrices which admit non-real left eigenvalues. For three by three case we review previous work by Tevian Dray and Corinne Manogue of real eigenvalue problem and study characteristic equations which admit non-real roots that are correspond to non-real left eigenvalues. Finally, we discuss the right spectrum using the associator method, and provide examples using "pyoctonion" python library. Interesting applications and open problems for future studies in this literature are also included.

In this thesis we classify singular vectors in scalar parabolic Verma modules for those pairs (sl(n, C), p) of complex Lie algebras where the homogeneous space SL(n, C)/P is the Grassmannian of k-planes in Cn . We calculate cohomology of nilpotent radicals with values in certain unitarizable highest weight modules. According to [BH09] these modules have BGG resolutions with weights determined by this cohomology. Such resolutions induce complexes of invariant differential operators on sections of associated bundles over Hermitian symmetric spaces. We describe formal completions of unitarizable highest weight modules that one can use to modify method from [CD01] that constructs sequences of differential operators over any 1-graded (aka almost Hermitian) geometry. We suggest uniform description of octonionic planes that could serve as a basis for better understanding of the exceptional Hermitian symmetric space for group E6.