Conditions for positive definiteness: matrix inequality

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 $C-B^T A^{-1} B \succ 0.$