The aim of this paper is to study the matrix discrepancy problem. Assume that $ξ_1,\ldots,ξ_n$ are independent scalar random variables with finite support and $\mathbf{u}_1,\ldots,\mathbf{u}_n\in \mathbb{C}^d$. Let $\mathcal{C}_0$ be the minimal constant for which the following holds: \[ {\rm Disc}(\mathbf{u}_1\mathbf{u}_1^*,\ldots,\mathbf{u}_n\mathbf{u}_n^*; ξ_1,\ldots,ξ_n)\,\,:=\,\,\min_{\varepsilon_1\in \mathcal{S}_1,\ldots,\varepsilon_n\in \mathcal{S}_n}\bigg\|\sum_{i=1}^n\mathbb{E}[ξ_i]\mathbf{u}_i\mathbf{u}_i^*-\sum_{i=1}^n\varepsilon_i\mathbf{u}_i\mathbf{u}_i^*\bigg\|\leq \mathcal{C}_0\cdotσ, \] where $σ^2 = \big\|\sum_{i=1}^n \mathbf{Var}[ξ_i](\mathbf{u}_i\mathbf{u}_i^*)^2\big\|$ and $\mathcal{S}_j$ denotes the support of $ξ_j, j=1,\ldots,n$. Motivated by the technology developed by Bownik, Casazza, Marcus, and Speegle, we prove $\mathcal{C}_0\leq 3$. This improves Kyng, Luh and Song's method with which $\mathcal{C}_0\leq 4$. For the case where $\{\mathbf{u}_i\}_{i=1}^n\subset \mathbb{C}^d$ is a unit-norm tight frame with $ n\leq 2d-1$ and $ξ_1,\ldots,ξ_n$ are independent Rademacher random variables, we present the exact value of ${\rm Disc}(\mathbf{u}_1\mathbf{u}_1^*,\ldots,\mathbf{u}_n\mathbf{u}_n^*; ξ_1,\ldots,ξ_n)=\sqrt{\frac{n}{d}}\cdotσ$, which implies $\mathcal{C}_0\geq \sqrt{2}$.