This post is about the eigenvalues and singular values of perturbed matrices.

Eigenvalues of perturbed Hermitian matrices.

It is well known that the sum of two positive definite matrices are sill positive definite. Weyl’s inequality gives an estimation of the eigenvalues of their sum as follows: Let \( A, B, C \in \mathbb{C}^{n \times n} \) be Hermitian matrices with \( A = B+C \). Suppose that there eigenvalues are \(a_i\), \(b_i\), \(c_i\), \( i \in [n] \) and are in a descent order, that is \(a_1 \geq a_2 \geq \ldots \geq a_n\) and so on. Then \[ b_{j} + c_k \leq a_i \leq b_s + c_t \quad \text{for } j+k - n \geq i \geq s+t-1. \] In particular, we have that \(b_i + c_n \leq a_i \leq b_i + c_1\) for \( i \in [n] \).

Fails in non-Hermitian case

Weyl’s inequality does not holds for non-Hermitian matrices. Consider the following example: \[ B = \begin{bmatrix}1 & 3\\ 0 & 1 \end{bmatrix}, C = \begin{bmatrix}1 & 0\\ 3 & 1 \end{bmatrix}.\] Note that the eigenvalues of \( B \) and \( C \) are all positive. If Weyl’s inequality holds, then the eigenvalues of \( B + C \) are supposed to be positive. However, \( \det(B + C) < 0 \).

Singular values of perturbed matrices.

The following results come from Corollary 7.3.5 of (Horn and Johnson 2012). Let \(A, B \in \mathbb{R}^{n \times r}\), where $ r ≤ n $ . Let \(\sigma_1(A) \geq \cdots \geq \sigma_r(A)\) and \(\sigma_1(B) \geq \cdots \geq \sigma_r(B)\) be the non-increasingly ordered singular values of \(A\) and \(B\), respectively. Then

1. \(\left|\sigma_i(A)-\sigma_i(B)\right| \leq\|A-B\|_{2}\) for each \(i=1, \ldots, q\);
2. \(\sum_{i=1}^q\left(\sigma_i(A)-\sigma_i(B)\right)^2 \leq\|A-B\|_F^2\).

References