basic matrix inequalities

Let $<$ denote the matrix order (Loewner order). Most the following inequalities apply to more general linear operators.

As you’d expect, there are matrix versions of the Markov and Chebyshev inequalities. A good overview is given in Appendix C here: https://arxiv.org/pdf/quant-ph/0012127

For more sophisticated matrix inequalities (which often use the following inequalities in the background) see matrix inequalities.

Markov

For matrix $X$ and PSD $A$,

$$\mathbb{P}(X\nleqq A)\le \text{Tr}(\mathbb{E}[X]A^{-1}).$$

Of course, this reduces to usual Markov inequality (basic inequalities:Markov’s inequality).

Chebyshev

Markov’s inequality extends to Chebyshev’s inequality in the same way as in the scalar case:

$$\mathbb{P}(|X-\mathbb{E}X|\nleqq A)\le \text{Tr}(\mathbb{E}|X-\mathbb{E}X{|}^2{A}^{-2}).$$

A Chernoff-like inequality

For matrix $Y$, symmetric matrix $B$ and matrix $T$ such that $T^{\ast }T>0$ where $T^{\ast }$ is the conjugate transpose of $T$, we have

$$\mathbb{P}(Y\nleqq B)\le \text{Tr}(\mathbb{E}\exp (TY{T}^{\ast }-TB{T}^{\ast })).$$

We can prove this easily using Markov’s inequality:

$$\begin{array}{rl}\mathbb{P}(Y\nleqq B)& =\mathbb{P}(Y-B\nleqq 0)\\ & =\mathbb{P}(TY{T}^{\ast }-TB{T}^{\ast }\nleqq 0)\\ & =\mathbb{P}(\exp (TY{T}^{\ast }-TB{T}^{\ast })\nleqq I)\\ & \le \text{Tr}(\mathbb{E}\exp (TY{T}^{\ast }-TB{T}^{\ast })I^{-1}).\end{array}$$

Here we’ve used that the exponential of the zero matrix is the identity. Note also that since the trace is a linear operator, so

$$\text{Tr}(\mathbb{E}\exp (TY{T}^{\ast }-TB{T}^{\ast }))=\mathbb{E}\text{Tr}(\exp (TY{T}^{\ast }-TB{T}^{\ast })).$$