The transmission matrix is a unique tool to control light through a scattering medium. A monochromatic transmission matrix does not allow temporal control of broadband light. Conversely, measuring multiple transmission matrices with spectral resolution allows fine temporal control when a pulse is temporally broadened upon multiple scattering, but requires very long measurement time. Here, we show that a single linear operator, measured for a broadband pulse with a co-propagating reference, naturally allows for spatial focusing, and interestingly generates a two-fold temporal recompression at the focus, compared with the natural temporal broadening. This is particularly relevant for non-linear imaging techniques in biological tissues.When monochromatic coherent light propagates in a medium with high refractive index inhomogeneities, it quickly develops into a speckle. Despite the complex structure of speckle patterns, each speckle grain has a deterministic relation to the input fields [1]. Over the last decade, wavefront shaping has turned to be an efficient tool to control monochromatic light through highly scattering systems [2], notably by exploiting the transmission matrix [3].Under illumination with a source of large bandwidth, each spectral component can generate a different speckle pattern [4]. Therefore one needs to adjust these additional spectral/temporal degrees of freedom to temporally control the output pulse [5]. This can be achieved using methods such as nonlinear optical processes [6,7], time-gating [8,9], and frequency-resolved measurements [10,11]. However, these methods underly either low signal-to-noise measurements (non-linear processes), or stability issues as they require lengthy acquisition procedures [12] and the need of external reference.An alternative approach is to use self-referencing signals, at the expense of lacking control on spectral degrees of freedom. Recently, "broadband wavefront shaping" experiments reported outcomes disparate from what is expected from monochromatic wavefront shaping, such as a decrease in the independent spectral degrees of freedom [13,14], and recovery of pure polarization states [15]. Notably, these results could have an impact on biomedical imaging [16]. Nonetheless, the exact temporal properties of the obtained output pulse via broadband wavefront shaping remain elusive. In this letter, we report the first characterization of the so-called broadband transmission matrix of a scattering medium. We exploit it for focusing purposes, and we analyze and interpret its temporal behavior. Unexpectedly, the characterized average pulse length is shorter than the pulse propagating without shaping. Figure 1a illustrates and summarizes propagation of broadband light (ultrashort pulse of duration δt, spectral width ∆λ) through an optically thick scattering medium. Transmitted light results in a speckle intensity pattern with low contrast C 0 < 1. This low contrast results from the incoherent summation of various uncorrelated speckles corresponding to different spect...