Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.

A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure.

A Łojasiewicz-type estimate is a powerful tool in studying the rigidity properties of the harmonic map heat flow. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to $\mathbb{S}^{2}$ ($J. Differential Geom.$$45$(1997), 593–610). In this note, using an inequality due to Sobolev, we will derive the same estimate for maps from $\mathbb{S}^{2}$ to a compact Kähler manifold$N$with nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to$N$, which generalizes Topping’s result.

We describe$T$-equivariant Schubert calculus on$G$($k$,$n$),$T$being an$n$-dimensional torus, through derivations on the exterior algebra of a free$A$-module of rank$n$, where$A$is the$T$-equivariant cohomology of a point. In particular,$T$-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Giambelli and determinantal restriction formulas for the Grassmannian,$Pure Appl. Math. Quart.$$2$(2006), 699–717).

We show that the transition probability of the Markov chain ($G$($i$,1),...,$G$($i$,$n$))_{$i$≥1}, where the$G$($i$,$j$)’s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for$G$($m$,$n$). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.