Determinants of Positive Definite Matrices: Positivity, the Bound $\log\det A\le\mathrm{tr}\,A-d$, Bounds under Pinching, and the Expansion of $\det(I+tB)$

In the setting of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation, let $d$ be a natural number with $1\le d$, read in $\mathbb{R}$ as in The Real Numbers: Standing Notation and Background §numbers where a real number is required. For a real $d\times d$ matrix $A$, $\det A$ is its determinant and $\mathrm{tr}\,A$ its trace; positive definiteness is that of that definition, and $\log$ is the natural logarithm.

1. (Positivity)¶ If $A\in\mathcal{S}(d)$ is positive definite, then $0<\det A$.

2. (Logarithm of the determinant)¶ If $A\in\mathcal{S}(d)$ is positive definite, then

3. (Pinching)¶ Let $\varepsilon$ and $L$ be positive real numbers and let $A\in\mathcal{S}(d)$ satisfy $\varepsilon I_{d}\preceq A\preceq L\,I_{d}$. Then $A$ is positive definite and

4. (Expansion of $\det(I_{d}+tB)$)¶ There are nonnegative real numbers $C_{d}$ and $K_{d}$ and a positive real number $c_{d}\le1$, depending only on $d$, with the following property. Let $B$ be a real $d\times d$ matrix, let $m$ be a nonnegative real number with $|B_{ij}|\le m$ for all $i,j\in[d]$, and let $t\in\mathbb{R}$ satisfy $|t|\,m\le1$. Then

and if moreover $|t|\,m\le c_{d}$, then $0<\det(I_{d}+tB)$ and