A Jordan block is one eigenvalue plus one nilpotent shift

Key idea

A size \(k\) Jordan block is \(J_k(\lambda)=\lambda I+N\), where \(N\) has \(1\) just above the diagonal. The nilpotent part satisfies \(N^k=0\) but \(N^{k-1}≠0\). A nontrivial block, meaning \(k>1\), is the basic obstruction to diagonalization.

Recognition checklist

• Diagonal entries give the eigenvalue \(\lambda\), repeated \(k\) times.
• The superdiagonal \(1\) entries create the chain relation.
• \((J_k(\lambda)-\lambda I)^k=0\).
• A single block contributes exactly one ordinary eigenvector direction.
• For a nontrivial \(2\times2\) block, \(J-\lambda I\) has rank \(1\) and kernel dimension \(1\).
• A block of size \(1\) is already diagonal.

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A chain explains where generalized eigenvectors sit

Key idea

Let \(N=A-\lambda I\). A length \(3\) chain \(v_1,v_2,v_3\) satisfies \(Nv_1=0\), \(Nv_2=v_1\), and \(Nv_3=v_2\). In the ordered basis \((v_1,v_2,v_3)\), the restriction of \(A\) has one Jordan block \(J_3(\lambda)\).

Chain recipe

• Start with an ordinary eigenvector \(v_1\in\ker N\).
• Find \(v_2\) with \(Nv_2=v_1\) when a length \(2\) chain exists.
• Find \(v_3\) with \(Nv_3=v_2\) for a length \(3\) chain.
• The top vector \(v_k\) is killed by \(N^k\), but not by \(N^{k-1}\).
• Each chain contributes one Jordan block.

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Block counts separate algebraic from geometric multiplicity

Key idea

For a fixed eigenvalue \(\lambda\), algebraic multiplicity is the total size of all \(\lambda\)-blocks. Geometric multiplicity is \(\dim\ker(A-\lambda I)\), which equals the number of \(\lambda\)-blocks because each block contributes one eigenvector direction.

Multiplicity facts

• One block of size \(3\): algebraic multiplicity \(3\), geometric multiplicity \(1\).
• Blocks of sizes \(2\) and \(1\): algebraic multiplicity \(3\), geometric multiplicity \(2\).
• Three blocks of size \(1\): algebraic multiplicity \(3\), geometric multiplicity \(3\), so the \(\lambda\)-part is diagonal.
• For each eigenvalue, diagonalizability requires geometric multiplicity equal to algebraic multiplicity.

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The minimal polynomial sees the largest block

Key idea

If the largest Jordan block for \(\lambda\) has size \(s\), then \(m_A(X)\) contains the factor \((X-\lambda)^s\). The characteristic polynomial counts total block size; the minimal polynomial records the largest block size for each eigenvalue.

Minimal polynomial rules

• A single \(3\times3\) block for \(\lambda\) has minimal polynomial \((X-\lambda)^3\).
• The scalar matrix \(\lambda I\) has minimal polynomial \(X-\lambda\).
• If the minimal polynomial has no repeated factor and splits into linear factors, every block has size \(1\).
• Repeated factors in \(m_A\) identify nontrivial Jordan blocks.

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Generalized eigenspaces are stable kernels of powers

Key idea

The generalized eigenspace for \(\lambda\) is the eventual kernel of powers of \(N=A-\lambda I\): \(\ker N\subseteq\ker N^2\subseteq\cdots\). In finite dimension this chain stabilizes, and the stable space contains all vectors in the Jordan chains for \(\lambda\).

Kernel ladder

• \(\ker N\) is the ordinary eigenspace.
• \(\ker N^2\) adds vectors at height at most \(2\) in chains.
• \(\ker N^k\) adds vectors at height at most \(k\).
• For a fixed eigenvalue, the stabilized dimension equals the algebraic multiplicity.
• Different eigenvalues give generalized eigenspaces whose direct sum is the whole space when the characteristic polynomial splits.

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Nilpotent blocks make trace, determinant, and powers easy

Key idea

A nilpotent Jordan block is \(J_k(0)\). Its only eigenvalue is \(0\), and its nilpotency index is \(k\). More generally, Jordan matrices are triangular, so trace and determinant come from diagonal eigenvalues, counted with algebraic multiplicity.

Invariant shortcuts

• A nilpotent block of size \(k\) satisfies \(N^k=0\) and \(N^{k-1}≠0\).
• The nilpotency index of a nilpotent matrix is the largest nilpotent block size.
• The trace of a Jordan matrix is the sum of diagonal eigenvalues.
• The determinant of a Jordan matrix is the product of diagonal eigenvalues.
• A size \(3\) block with eigenvalue \(\lambda\) has trace \(3\lambda\).

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Most mistakes confuse blocks, fields, and diagonalizability

Common traps

• Algebraic versus geometric: add block sizes for algebraic multiplicity, count blocks for geometric multiplicity.
• Largest versus total: the minimal polynomial sees the largest block, not the sum of block sizes.
• Generalized versus ordinary: \(Nv=0\) is ordinary; \(N^kv=0\) for some larger \(k\) is generalized.
• Diagonalizable: every block must have size \(1\); one block of size \(2\) is enough to fail.
• Field: full Jordan form may require working over \(\mathbb{C}\) when eigenvalues are not in the original field.
• Orthogonality: Jordan form is a similarity normal form, not usually an orthogonal diagonalization.

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Final recap

• A generalized eigenvector satisfies \((A-\lambda I)^k v=0\) for some \(k\ge1\).
• A Jordan block is \(\lambda I\) plus a nilpotent shift.
• A length \(k\) Jordan chain produces one block of size \(k\).
• Algebraic multiplicity is total block size.
• Geometric multiplicity is the number of blocks.
• The largest block size is the exponent in the minimal polynomial.
• Diagonalizable means every Jordan block has size \(1\).
• The generalized eigenspace is the stabilized kernel of powers of \(A-\lambda I\).
• Nilpotency index is the largest nilpotent block size.
• Trace and determinant of a Jordan matrix come from the diagonal entries.