An eigenvector is the bottom of a possible chain. For N=[[0,1,0],[0,0,1],[0,0,0]], the standard vectors satisfy Ne₁=0, Ne₂=e₁ and Ne₃=e₂. Starting at e₃, successive applications descend the chain until zero. For T=4I+N these same equations become (T−4I)e₂=e₁ and (T−4I)e₃=e₂. The operator has only one independent eigenvector, yet the three chain vectors form a basis.
Predict
Predict how many dimensions enter ker Nᵏ at each step. In a chain of length s, exactly min(k,s) chain vectors are killed by Nᵏ. Therefore the increment dim ker Nᵏ−dim ker Nᵏ⁻¹ counts chains of length at least k. These increments must decrease or stay constant. A claimed nilpotent nullity list with increasing increments is impossible, and this gives a useful diagnosis before any basis construction.
Experiment
Use two chains of lengths four and one. Construct a table of k=0,1,2,3,4 and nullities 0,2,3,4,5. The increments 2,1,1,1 show two blocks altogether, only one reaching length two, and that same block reaching length four. Reverse the process for nullities 0,3,5,6,6: increments 3,2,1,0 give block sizes 3,2,1. Draw each chain as a row of boxes pointing toward zero and mark which boxes are killed at each power.
Understand
Here is a chain-basis existence argument for nilpotent N, by induction on dimension. Let W=im N, a proper invariant subspace unless V is zero. By induction W has chain bases. For each top vector wᵢ of a chain in W, choose vᵢ with Nvᵢ=wᵢ and extend that chain upward. The classes of the tops wᵢ form a basis of W/NW. If a combination of the vᵢ lies in W, applying N puts the same combination of wᵢ in NW, forcing all coefficients zero. Thus the extended chains are independent. Their span E contains W and has N(E)=W. Every v∈V differs from some e∈E by a vector in ker N; add a basis of a complement of E∩ker N within ker N as length-one chains. This completes a chain basis of V.
Apply
When the characteristic polynomial splits, apply that nilpotent construction to each generalized eigenspace for T, replacing N by T−λI. Ordering each chain from bottom to top gives a Jordan block with λ on its diagonal and ones on its superdiagonal. Kernel increments recover the number of blocks of each length, proving uniqueness of the block multiset for each eigenvalue. The basis itself is highly nonunique. For powers, commuting λI and N allow the binomial formula; a length-two block gives (λI+N)ᵏ=λᵏI+kλᵏ⁻¹N when λ≠0 and k≥1. The polynomial factor k explains transient behavior invisible in eigenvalues alone.
Continue
Jordan form is an exact structural theorem, not usually a stable floating-point algorithm. Tiny perturbations can split repeated eigenvalues and change block sizes. The kernel-rank calculations used to identify blocks must therefore be interpreted cautiously with noisy data. Next, inner products add geometric structure and eventually give orthonormal decompositions for special operator classes; those decompositions have much better numerical behavior than arbitrary Jordan bases.
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A nilpotent operator has nullities dim ker N=3, dim ker N²=5, dim ker N³=6, dim ker N⁴=6. How many blocks have size exactly 2?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For those nullities, what is the largest Jordan block size?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For N=[[0,2,0],[0,0,3],[0,0,0]], construct a chain v₁,v₂,v₃ with Nv₁=0,Nv₂=v₁,Nv₃=v₂.
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Recover nilpotent block sizes from nullities 2,4,5,6,6 at powers 1 through 5.
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For J=λI+N with N³=0 and λ≠0, derive Jᵏ for integer k≥2.
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Explain both the existence hypothesis and the uniqueness meaning of Jordan form.
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Reflect and continue
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