Conditions for positive definiteness: matrix inequality

Question

Let $0<\alpha<1$ and $A,B\in\mathbb{R}^{n\times n}$. I am trying to find conditions on $A$ and $B$ such that \begin{equation} I_n-\frac{1}{\alpha}B^{\rm T}B-\frac{1}{4\alpha(1-\alpha)}( A^{\rm T}B+A)^{\rm T}( A^{\rm T}B+A)>0. \end{equation} However, I do not know how to proceed. Any idea or suggestion is appreciated.

Answer 1

The problem reduces to showing that \begin{equation} \lambda_{\max}\left(\frac{1}{\alpha}B^{\rm T}B+\frac{1}{4\alpha(1-\alpha)}( A^{\rm T}B+A)^{\rm T}( A^{\rm T}B+A)\right)<1. \end{equation} Let $\sigma_A$ and $\sigma_B$ be the largest singular values of $A$ and $B$, respectively. Then, it follows that \begin{align} \lambda_{\max}&\left(\frac{1}{\alpha}B^{\rm T}B+\frac{1}{4\alpha(1-\alpha)}( A^{\rm T}B+A)^{\rm T}( A^{\rm T}B+A)\right)\\&\leq\frac{1}{\alpha}\sigma_B^2+\frac{1}{4\alpha(1-\alpha)}\sigma_{\max}^2\left(A^{\rm T}B+A\right)\\ &\leq\frac{1}{\alpha}\sigma_B^2+\frac{1}{4\alpha(1-\alpha)}\left(\sigma_{\max}^2\left(A^{\rm T}B\right)+2\sigma_{\max}\left(A^{\rm T}B\right)\sigma_A+\sigma_A^2\right)\\ &\leq\frac{1}{\alpha}\sigma_B^2+\frac{1}{4\alpha(1-\alpha)}\left(\sigma_A^2\sigma_B^2+2\sigma_A^2\sigma_B+\sigma_A^2\right)\\ \end{align} Next, assume \begin{equation} \frac{1}{\alpha}\sigma_B^2+\frac{1}{4\alpha(1-\alpha)}\left(\sigma_A^2\sigma_B^2+2\sigma_A^2\sigma_B+\sigma_A^2\right)<1, \end{equation} and it follows that \begin{equation} 4\alpha^2-4\alpha(1+\sigma_B^2)+4\sigma_B^2+\sigma_A^2(\sigma_B+1)^2<0, \end{equation} which implies $\sigma_B<1$ and $\sigma_A<1-\sigma_B$ since $0<\alpha<1$. Thus, $\sigma_A+\sigma_B<1$ is a sufficient condition.

Answer 2

Hint:

Rewrite it as should be positive definite as you desired:

$$Sc \triangleq \Big (I_n - \frac{1}{\alpha}B^T B \Big )- \frac{1}{4\alpha (1-\alpha)}\Big\{(A^TB+A)^T I_n (A^TB+A)\Big \} \succ 0$$

Then, by using Schur complement, $S_c$ is positive definite if and only if the matrix $S$ defined as:

$$S \triangleq \begin{pmatrix} \frac{1}{4\alpha (1-\alpha)}I_n & A^TB+A \\ (A^TB+A)^T & I_n - \frac{1}{\alpha}B^T B \end{pmatrix},$$

is:

$$S \succ 0 \quad \text{with the conditions mentioned below would be satisfied.} $$ Also, we have:

$$ \lambda_i(S) >0; \quad \text{for all } i = 0,1, ..., n.$$

Now, it needs to check for which $A$ and $B$ the above relationship (Schur complement) can be satisfied.

Second, is to show the eigenvalues of such block matrix is positive.

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Side note: Schur complement

For any symmetric matrix X, of the form:

$$X \triangleq \begin{pmatrix} A & B \\ B^T & C \end{pmatrix},$$

if $A$ is invertible, then the following statement holds:

$X \succ 0$ if and only if $A \succ 0$ and $C-B^T A^{-1} B \succ 0.$