This paper is concerned with the largest eigenvalue of the Wishart-type random matrix <inline-formula> <tex-math notation="LaTeX">\mathbf {{W}}=\mathbf {{X}}\mathbf {{X}}^\dagger</tex-math></inline-formula> (or <inline-formula><tex-math notation="LaTeX">\mathbf {{W}}=\mathbf {{X}}^\dagger \mathbf {{X}}</tex-math> </inline-formula>), where <inline-formula><tex-math notation="LaTeX">\mathbf {{X}}</tex-math></inline-formula> is a complex Gaussian matrix with unequal variances in the real and imaginary parts of its entries, i.e., <inline-formula> <tex-math notation="LaTeX">\mathbf {X}</tex-math></inline-formula> belongs to the noncircularly symmetric Gaussian subclass. By establishing a novel connection with the well-known complex Wishart ensemble, we here derive exact and asymptotic expressions for the largest eigenvalue distribution of <inline-formula><tex-math notation="LaTeX">\mathbf {{W}}</tex-math></inline-formula>, which provide new insights on the effect of the real-imaginary variance imbalance of the entries of <inline-formula><tex-math notation="LaTeX">\mathbf {X}</tex-math></inline-formula>. These new results are then leveraged to analyze the outage performance of multiantenna systems with maximal ratio combining subject to Nakagami-<inline-formula><tex-math notation="LaTeX">q</tex-math></inline-formula> (Hoyt) fading.