Jordan Normal Form 1 | Overview [dark version]

Jordan normal form is the universal replacement for diagonalization: every square matrix with complex entries can be converted—via a similarity transform—into a structured block matrix that makes powers and other computations tractable even when diagonalization fails. When a matrix is diagonalizable, its Jordan normal form collapses to a diagonal matrix. But in the general case, the “diagonal” picture is replaced by Jordan blocks, and the arrangement of those blocks is what captures the matrix’s deeper algebraic structure.

The starting point is the familiar diagonalizable case. A matrix A is diagonalizable if an invertible matrix X exists such that X^{-1}AX becomes a diagonal matrix D. That decomposition is powerful because computing A^n reduces to computing D^n, which is easy since it just raises diagonal entries. The natural question is what to do when diagonalization is impossible. Jordan normal form answers it by guaranteeing that for any square matrix A over the complex numbers, there exists an invertible X such that X^{-1}AX equals a Jordan matrix J. Two caveats matter immediately: J is not unique in general (different Jordan forms can be similar), and even if A has only real entries, J may still require complex numbers.

To make the structure concrete, the discussion uses a 9×9 example with eigenvalues 2, 3, and 4. Their algebraic multiplicities are chosen as 3, 2, and 4 respectively, summing to 9. Algebraic multiplicity is defined through the characteristic polynomial: it counts how many times an eigenvalue appears as a root. In a diagonalizable situation, each eigenvalue would simply occupy the diagonal with its multiplicity. Jordan normal form instead groups equal eigenvalues into Jordan blocks. The key rule introduced is that the algebraic multiplicity determines the total size of the Jordan blocks for that eigenvalue.

What happens inside a Jordan block depends on geometric multiplicity, which equals the dimension of the kernel of (A − λI). For a 3×3 Jordan block (algebraic multiplicity 3), there are three possibilities. If geometric multiplicity is 3, the block splits into three 1×1 Jordan boxes (no ones above the diagonal). If geometric multiplicity is 2, the block breaks into two boxes: one 2×2 block and one 1×1 block, with ones above the diagonal inside the 2×2 part. If geometric multiplicity is 1, the entire 3×3 block is a single Jordan block, forcing ones on the superdiagonal.

For larger blocks, algebraic and geometric multiplicities no longer always pin down the exact block sizes. In a 4×4 block with geometric multiplicity 2, two distinct decompositions are possible: either two 2×2 blocks, or one 3×3 block plus one 1×1 block. That distinction signals when more computation is needed beyond multiplicities.

Finally, a recipe for building the Jordan normal form is laid out. For each eigenvalue λ, compute its algebraic multiplicity from the characteristic polynomial, compute geometric multiplicity as dim ker(A − λI), and—when block structure remains ambiguous—compute kernels of higher powers of (A − λI)^k. The process continues until the kernel dimensions stop changing. The construction of the similarity matrix X is treated as an additional step reserved for a later, detailed example.