Rank-One Perturbation Theory

• Rank-One Perturbation Theory is the study of how adding a single rank update alters a matrix or operator's spectrum via characteristic polynomials and Jordan block nuances.
• The analysis uses necessary and sufficient conditions based on matching algebraic multiplicities with the size of the largest Jordan block, ensuring precise spectral assignments.
• Its practical implications extend to control theory, random matrix theory, and signal processing, facilitating stability analysis under minimal perturbations.

Rank-one perturbation theory concerns the spectral and structural changes induced on matrices and operators by adding a perturbation of rank one. The classical and modern developments of this theory highlight the interplay between algebraic invariants (characteristic polynomials, Jordan block sizes, cyclicity), operator-theoretic resolvent identities, and connections to key areas such as random matrix theory, control, and mathematical physics. The spectral consequences are sharpest when the ambient field is algebraically closed and full Jordan decomposition is available.

1. Structural Framework: Notation, Definitions, and Spectral Data

Let $A \in F^{n \times n}$ be a matrix over an algebraically closed field $F$. The canonical spectral invariant is the characteristic polynomial $p_A(t) = \det(tI - A)$, monic of degree $n$. For any monic $q(t) \in F[t]$ of degree $n$, fundamental questions arise: which $q(t)$ can appear as the characteristic polynomial of a rank one perturbation $A+B$, i.e., $B = uv^T$ for $u,v \in F^n$?

Key quantities for each eigenvalue $\lambda \in F$:

• Algebraic multiplicity: $$\operatorname{alg}_\lambda(A) = m_\lambda(p_A) = \text{multiplicity of } (t-\lambda)$$ in $p_A(t)$.
• Jordan block size: $j_\lambda(A) =$ size of largest Jordan block for $\lambda$.

Simultaneously, the class of rank-one matrices forms the set $\{ uv^T : u,v \in F^n \}$, and the rank-one perturbation problem is nontrivial due to its effect on the geometric and algebraic multiplicities of eigenvalues.

2. Main Theorem and Necessary-Sufficient Condition for Spectral Assignability

Merzel–Mináč–Muller–Pasini–Nguyen's principal result provides a complete criterion for the existence of $B$ such that $p_{A+B}(t) = q(t)$ (Merzel et al., 2021): Main Theorem: There exists $B$ of rank one with $p_{A+B}(t) = q(t)$ if and only if, for all $\lambda \in F$,

$$m_\lambda(q) \geq \operatorname{alg}_\lambda(A) - j_\lambda(A)$$

This logarithmic-type inequality is both necessary and sufficient: it quantifies exactly how many roots at $\lambda$ must persist in $q(t)$, depending on the initial multiplicities and largest Jordan block.

The sufficiency leverages explicit construction in Jordan canonical coordinates, exploiting the block structure.

3. Proof Strategy, Weinstein–Aronszajn Identity, and Construction Principles

The analysis proceeds by:

• Necessity: Via rank estimates, for any rank-one $B$ and Jordan block $J_{\lambda,n}$, the algebraic multiplicity at $\lambda$ cannot decrease by more than the size of the largest block: $$m_\lambda(q)\geq \operatorname{alg}_\lambda(A) - j_\lambda(A)$$.
• Sufficiency: For the Jordan decomposition $A = \bigoplus_{\lambda} A_{\lambda}$, a rank-one perturbation is constructed as follows:
  1. For a single block $J_{\lambda,n}$, any monic polynomial is attainable, since $j_\lambda(A)=n$ and the constraint vacuously holds. With the expansion

  $$(tI-J_{\lambda,n})^{-1} = \sum_{i=0}^{n-1} (t-\lambda)^{-i-1} N^i$$

  one can tune $u,v$ so that $v^T (tI-A)^{-1} u = [p_A(t) - q(t)]/p_A(t)$ matches the desired rational function. 2. For direct sums of distinct blocks: the Chinese remainder theorem in $F[t]/(p_A)$ allows decomposition into summands supported on each block; rank-one updates are patched accordingly. 3. For repeated eigenvalues: a hybrid of steps 1 and 2 treats blocks of maximal size separately.

The pivotal formula for a rank-one update is the Weinstein–Aronszajn identity: $$\det(tI - (A + uv^T)) = p_A(t) \cdot [1 - v^T (tI - A)^{-1} u]$$ Thus,

$$p_{A+B}(t) = p_A(t)\bigl(1 - v^T (tI-A)^{-1} u\bigr)$$

The entire spectral assignment reduces to engineering $u,v$ so that $(1 - v^T (tI-A)^{-1}u)$ adjusts $p_A(t)$ to $q(t)$, subject to the necessary constraints.

4. Algebraic and Jordan-theoretic Consequences

Table: Critical invariants governing spectral assignability

Parameter	Meaning	Role in Theorem
$\operatorname{alg}_\lambda(A)$	Algebraic multiplicity at $\lambda$	Baseline count of roots at $\lambda$
$j_\lambda(A)$	Largest Jordan block at $\lambda$	Controls maximal loss of roots
$m_\lambda(q)$	Multiplicity of $(t-\lambda)$ in $q$	Target count of roots in $q$

Implications:

• Diagonalizable $A$: $j_\lambda(A) = 1$ for all $\lambda$; hence $m_\lambda(q)\geq \operatorname{alg}_\lambda(A)-1$. One can add or remove at most one root per eigenvalue.
• Single Jordan block: For $A = J_{\lambda,n}$, any $q(t)$ is realizable; the full spectrum can be arbitrarily assigned.
• Generic case: Repeated eigenvalues with variable block sizes require careful partitioning and the possibility of spectral ‘loss’ limited by the block structure.

5. Illustrative Examples and Special Cases

• Zero-matrix case: For $A = 0_{2}$, both $\operatorname{alg}_0(A)=2$ and $j_0(A)=2$, so any quadratic monic $q(t)$ is obtainable (no constraint); explicit construction given in the paper.
• Complex field $F = \mathbb{C}$: The result recovers classical theorems (Lidskiĭ, Kato) on eigenvalue shifts for small rank-one perturbations; isolated eigenvalues can be targeted within the constraint.
• Direct computation for diagonal $A$: For $A=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$,

$$\frac{q(t)}{p_A(t)} = 1 - \sum_{i=1}^n \frac{\alpha_i}{t-\lambda_i}$$

Choosing all but one $\alpha_i$ zero gives a rank-one update.

6. Connections, Generalizations, and Implications

This theorem provides:

• A sharp improvement over earlier sufficient conditions (e.g., Cheung–Ng, Krupnik), by producing an explicit necessary-and-sufficient inequality involving both algebraic and geometric multiplicities.
• A unification of various spectral assignment problems, reducing even complicated spectral rearrangement to a single rank-one perturbation provided the Jordan structure is favorable.
• A foundational framework for understanding stability under low-rank updates in control, signal processing, and random matrix theory, where perturbations naturally arise as data or feedback are introduced.

The structure of permissible characteristic polynomials under rank-one perturbation is fully governed by the interaction between the algebraic multiplicities and the sizes of Jordan blocks, providing both an explicit algorithm for construction and a transparent obstruction theory for spectral assignment.

7. Further Technical Lemmas and Computational Identities

Key Lemmas:

• Weinstein–Aronszajn formula for block matrices: If $M$ and $N$ are $m \times n$ and $n \times m$ matrices,

$\det(I_m - MN) = \det(I_n - NM)$

Used repeatedly to pass between perturbation representations.

• Rank estimates: For arbitrary $X,Y$ of size $n$, and $k \geq 1$,

$$\operatorname{rank}((X+Y)^k) \leq k \cdot \operatorname{rank}(X) + \operatorname{rank}(Y^k)$$

For $Y$ of rank one, this controls multiplicities after perturbation.

These technical results illuminate why the main spectral assignment theorem is not only sharp but constructively optimal for rank-one updates.

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The theory as crystallized by Merzel–Mináč–Muller–Pasini–Nguyen (Merzel et al., 2021) thus closes the spectral perturbation problem for rank-one matrices, yielding a necessary and sufficient criterion for attainable characteristic polynomials under a minimal rank update and tightly linking the algebraic, geometric, and analytic invariants of the underlying operator.