Chapter 4: QR Decomposition and Numerical Rank. In the chapter mind map this icon labels Spectral Analysis & Shear/Projection Geometry. The discussion below is excerpted and lightly edited from § Theorem: Spectral Analysis of Rank-1 Perturbation in Mathematics for AI and Machine Learning. Related material also appears in Chapter 6 (Diagonalization $A=V\Lambda V^{-1}$), Chapter 7 (Spectral Theorem: $A = Q\Lambda Q^\top$).

For the rank-1 perturbation $T = I - \beta \mathbf u \mathbf u^\top$ where $\mathbf u$ is a unit vector:

• One eigenvalue: $\lambda_0 = 1 - \beta$ with eigenvector $\mathbf u$
• $d-1$ eigenvalues: $\lambda_i = 1$ for $i = 1, \ldots, d-1$ with eigenspace $\mathbf u^\perp$
• The eigenvector corresponding to $\lambda_0 = 1 - \beta$ is $\mathbf u$ (the direction of perturbation)
• The eigenvectors corresponding to $\lambda_i = 1$ span the hyperplane $\mathbf u^\perp$ (orthogonal to $\mathbf u$)

1. For $\mathbf u$: Therefore, $\mathbf u$ is an eigenvector with eigenvalue $1 - \beta$.

2. For any $\mathbf w \in \mathbf u^\perp$ (i.e., $\mathbf u^\top \mathbf w = 0$): Therefore, any vector orthogonal to $\mathbf u$ is an eigenvector with eigenvalue $1$.

3. Dimension count: Since $\mathbf u^\perp$ has dimension $d-1$ and $\mathbf u$ spans a 1-dimensional space, we have $d$ linearly independent eigenvectors, confirming the complete eigensystem.