MA Notes

Posted 2022-03-12 analysis-pde

Book: A. Figalli   《The Monge Ampere Equation and Its Application》

 

1.Let $A,B\in R^n\times n$, and assume that $A$ is invertible. Then,

$$\fracddt|_t=0det(A+tB)=det(A)tr(A^-1B)=tr(cof(A)^TB).$$

In addition, the latter formula holds also when $A$ is not invertible.

 

2.Let $A,B\in R^n\times n$, and assume that $A$ is invertible. Then,

$$\fracddt|_t=0det(A+tB)^-1=det(A)tr(A^-1B)=-A^-1BA^-1.$$

 

3.Let $A,B\in R^n\times n$ be symmetric nonnegative definite matrices. Then,

$$det(A+B)\geq det(A)+det(B),$$

$$det(A+B)^\frac1n\geq det(A)^\frac1n+det(B)^\frac1n.$$

Furthermore, if $A,B\in R^n\times n$ are symmetric positive definite matrices, then

$$\log det(\lambda A+(1-\lambda)B)\geq \lambda\log det(A) +(1-\lambda)\log det(B).$$

 

4. Given $A\in R^n\times n$, we denote its operator norm by $||A||$, i.e.,  $||A||:=\sup_|v|=1|Av|$.

Assume that there exists a constant $K>1$ such that  $\frac1KId\leq A^TA\leq AId$.

Then $||A||, ||A^-1||\leq \sqrtK$.

 

5. Area formula for the gradient of convex functions.

Let $\Omega$ be an open bounded set in $R^n$, and let $u:\Omega\rightarrow R$ be a convex function of class $C^1,1_loc$. Then,

$|\partial u(E)|=\int_E det（D^2u）dx , \forall E\subset \Omega Borel. $

 

6. Let $u: R^n\rightarrow R$ be a convex function, and assume that $u$ is affine on a line $\hatl$. Then  $\partial u(R^n)$ is containted inside a hyperplane orthogonal $\hatl$. In particular,  $|\partial u(R^n)|=0.$