Diagonalization and Jordan Form

GRE Math Subject · Axiom Academy

LESSON Diagonalization and Jordan Form Transforming matrices to simple, canonical forms and understanding non-diagonalizable cases A matrix is diagonalizable if there exists an invertible matrix such that: The columns of are eigenvectors of , and the diagonal entries of are the corresponding eigenvalues. Criterion: is diagonalizable if and only if for each eigenvalue : Find the characteristic polynomial Find all eigenvalues (roots of the polynomial) For each eigenvalue, solve to find a basis for the eigenspace Check: total number of basis vectors = (dimension of matrix). If not, matrix is not diagonalizable. Form with eigenvectors as columns, with eigenvalues on diagonal Useful fact: If all eigenvalues are distinct, the matrix is automatically diagonalizable. Diagonal matrices are easy to work with: is just the diagonal entries raised to the -th power Compute exponentials (differential equations) Every matrix can be put into Jordan normal form (even if not diagonalizable): A Jordan block for eigenvalue with size is: If all Jordan blocks are , then the matrix is diagonalizable. The GRE Subject Test emphasizes: Determining if a matrix is diagonalizable (compute multiplicities) Finding the diagonal form if diagonalizable Computing powers of matrices quickly using diagonalization Understanding Jordan blocks for defective matrices (basic structure only) Note: The exam rarely requires you to compute Jordan form explicitly. Focus on diagonalization and when it's possible.

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