Let J be a Jordan matrix. (a) Prove that J^k is a complete...

Let $J$ be a Jordan matrix. (a) Prove that $J^k$ is a complete matrix for some $k \geq 1$ if and only if either $J$ is diagonal, or $J$ is nilpotent with $J^k=0$. (b) Suppose that $A$ is an incomplete matrix such that $A^k$ is complete for some $k \geq 2$. Prove that $A^k=0$, and hence $A$ is nilpotent. (A simpler version of this problem appears in Exercise 8.3.8.)

Key Concepts

Jordan Matrix

A Jordan matrix is a block-diagonal matrix that is composed of Jordan blocks. Each block corresponds to an eigenvalue and has that eigenvalue on the diagonal with ones on the superdiagonal (and zeros elsewhere). This canonical form is very useful because it exposes the algebraic and geometric multiplicities of the eigenvalues and simplifies the study of functions of matrices, such as matrix powers.

Nilpotent Matrix

A nilpotent matrix is a square matrix for which some positive integer power equals the zero matrix. In the context of Jordan matrices, nilpotency arises when the only eigenvalue is zero. The structure of the Jordan blocks then guarantees that, as the power increases, the nonzero entries along the superdiagonals eventually vanish, leading to the zero matrix.

Diagonal Matrix

A diagonal matrix is a matrix where the entries outside the main diagonal are all zero. In the setting of Jordan matrices, a diagonal matrix is the special case where every Jordan block is 1×1. Diagonal matrices are well-behaved under matrix operations, such that raising a diagonal matrix to a power simply raises each diagonal entry to that power.

Matrix Exponentiation

Matrix exponentiation, or taking matrix powers, is an operation that can dramatically alter the structure of a matrix. For Jordan matrices, computing powers involves applying the binomial theorem to each Jordan block, which blends the diagonal eigenvalue powers with contributions from the nilpotent part. This results in behaviors that differ when the matrix is diagonal versus when it possesses nontrivial Jordan blocks.

Complete Matrix

A complete matrix, in this context, refers to a matrix that exhibits a certain fullness property—typically meaning that its nonzero entries are sufficiently arranged so that the matrix has maximal rank or can influence every entry when raised to a power. The question investigates conditions under which a Jordan matrix, or a general matrix, either retains or loses this complete structure upon exponentiation, linking it intrinsically to its underlying diagonal or nilpotent nature.

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