  Let $A$ be a $n\times n$ complex Hermitian matrix and let
$\lambda(A)=(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^n$ denote the
eigenvalues of $A$, counting multiplicities and arranged in non-increasing
order. Motivated by problems arising in the theory of low rank matrix
approximation, we study the spectral spread of $A$, denoted $\text{Spr}^+(A)$,
given by $\text{Spr}^+(A) =(\lambda_1-\lambda_{n}\, , \,
\lambda_2-\lambda_{n-1},\ldots, \lambda_{k}-\lambda_{n-k+1})\in \mathbb{R}^k$,
where $k=[n/2]$ (integer part). The spectral spread is a vector-valued measure
of dispersion of the spectrum of $A$, that allows one to obtain several
submajorization inequalities. In the present work we obtain inequalities that
are related to Tao's inequality for anti-diagonal blocks of positive
semidefinite matrices, Zhan's inequalities for the singular values of
differences of positive semidefinite matrices, extremal properties of direct
rotations between subspaces, generalized commutators and distances between
matrices in the unitary orbit of a Hermitian matrix.
