The interest in quantum information processing has given rise to the development of programming languages and tools that facilitate the design and simulation of quantum circuits. However, since the quantum theory is fundamentally based on linear algebra, these high-level languages partially hide the underlying structure of quantum systems. We show that in certain cases of practical interest, keeping a handle on the matrix representation of the quantum systems is a fruitful approach because it allows the use of powerful tools of linear algebra to better understand their behavior and to better implement simulation programs. We especially focus on the Joint EigenValue Decomposition (JEVD). After giving a theoretical description of this method, which aims at finding a common basis of eigenvectors of a set of matrices, we show how it can easily be implemented on a Matrix-oriented programming language, such as Matlab (or, equivalently, Octave). Then, through two examples taken from the quantum information domain (quantum search based on a quantum walk and quantum coding), we show that JEVD is a powerful tool both for elaborating new theoretical developments and for simulation.