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Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix. For real $t$, let us consider the one-parameter family $$ H(t) = H + t V$$ of Hermitian matrices. Kato's perturbation theory tells us that the eigenvalues $\lambda_k(t)$ and eigenfunctons $\phi_k(t)$ of this matrix-family can be chosen to beanalytic in $t$ and there is a family of unitary matrices $U(t)$ so that $\phi_k(t) = U(t)\phi_k(0)$.

Are there constants so that $$|\lambda_k(1)-\lambda_k(0)| \leq C_k ||V|| $$ $$||U(1)-U(0)|| \leq C ||V|| $$ holds true without assuming that all eigenvalues are simple? What are those constants?

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  • $\begingroup$ I'm unclear what the constant $C$ depends on in the second inequality. $\endgroup$
    – wlad
    Commented Oct 23, 2021 at 10:35
  • $\begingroup$ I'm unclear on how $U(t)$ is defined. What does $U$ depend on? Clearly not just $t$ $\endgroup$
    – wlad
    Commented Oct 23, 2021 at 10:36

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The eigenvalues are $1$-Lipschitz over the set of Hermitian matrices: $$|\lambda_k(B)-\lambda_k(A)\|\le\|B-A\|.$$ This Lipschitz property is a very different phenomenon than the analyticity of $t\mapsto\lambda_k(H+tV)$. Analyticity is false for $2$-parameter families when eigenvalues don't remain simple.

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