Final answer:
To answer the student's question, one must compute the determinant of (B - tI) to find the characteristic polynomial and its roots to determine the eigenvalues. Then construct the Jordan normal form based on the eigenvalues' algebraic and geometric multiplicities.
Step-by-step explanation:
The student has asked to find the characteristic polynomial P(t), the eigenvalues of a matrix B, and to determine the Jordan normal form J(B) for said matrix in the space of n x n complex matrices, M(n, C). To solve this, one would compute the determinant of the matrix (B - tI), where I is the identity matrix, to find the characteristic polynomial P(t). The roots of this polynomial are the eigenvalues. After finding the eigenvalues, the student would construct the Jordan normal form, which is a block diagonal matrix with Jordan blocks on the diagonal corresponding to each eigenvalue and its algebraic multiplicity.
To figure out the Jordan blocks, one might need to consider the geometric multiplicity of each eigenvalue and the structure of the generalized eigenspaces. However, without the specific matrix B provided, we cannot give the exact characteristic polynomial, eigenvalues, or Jordan normal form.