The joint linear complexity of multisequences is an important security measure for vectorized stream cipher systems. Extensive research has been carried out on the joint linear complexity of N-periodic multisequences using tools from Discrete Fourier transform. Each N-periodic multisequence can be identified with a single N-periodic sequence over an appropriate extension field. It has been demonstrated that the linear complexity of this sequence, the so called generalized joint linear complexity of the multisequence, may be considerably smaller than the joint linear complexity, which is not desirable for vectorized stream ciphers. Recently new methods have been developed and results of greater generality on the joint linear complexity of multisequences consisting of linear recurring sequences have been obtained. In this paper, using these new methods, we investigate the relations between the generalized joint linear complexity and the joint linear complexity of multisequences consisting of linear recurring sequences.