For AI agents: use /llms.txt for the Nakafa content index.
Even though matrix cannot be diagonalized, the following theorem still holds.
For matrix , if the characteristic polynomial of can be factored into linear factors, then is similar to an upper triangular matrix . The entries on the diagonal of this upper triangular matrix are all the eigenvalues of .
This similarity relationship is expressed as:
where is an invertible matrix.
This upper triangular form can be described more precisely through the Jordan Normal Form.
Let with characteristic polynomial:
with distinct eigenvalues . Then there exists an invertible matrix such that:
This form shows how a matrix can be organized into simpler blocks, where each block is associated with one particular eigenvalue.
For each , the Jordan block has a very distinctive structure. Imagine it like an almost perfect staircase, where each step has the same value (namely the eigenvalue ), but there are "connectors" in the form of the number at certain positions that make this structure unique.
In this structure, the eigenvalue dominates the main diagonal, while the number appears at certain positions above the diagonal (called the superdiagonal). The positions of and on the superdiagonal determine how the Jordan block is divided into smaller sub-blocks. This structure provides complete information about how the linear transformation operates on the vector space associated with that eigenvalue.
