This paper investigates the capacity of multi-antenna Gaussian channels in point-to-point communication systems. It considers a system with t transmit antennas and r receive antennas, where the received signal vector v depends on the transmitted vector u through the equation v = Hu + w, with H being the channel matrix and w being Gaussian noise. The capacity of the channel is given by the sum of logarithms of (1 + λ_i P_i / σ²), where λ_i are eigenvalues of HH†, and P_i is the waterfilling power allocation. The paper explores joint optimization of eigenvalues and power allocation to maximize channel capacity. It shows that when the signal-to-noise ratio is low, the optimal solution focuses power on a single channel, while at high signal-to-noise ratios, power is evenly distributed across multiple channels. In the pessimistic capacity analysis, the paper considers the worst-case scenario where an adversary chooses the eigenvalues. It finds that the worst-case capacity occurs when all eigenvalues are equal, while the best-case capacity is achieved with a single nonzero eigenvalue. The paper also analyzes two scenarios: one where the eigenvalues are known only at the receiver, and another where they are known at both ends. In the first case, the optimal power distribution is uniform, while in the second, the minimum achieving eigenvalues can take at most two values. The paper concludes that at low signal-to-noise ratios, the best channel has one nonzero eigenvalue, while the worst channel has all eigenvalues equal. At high signal-to-noise ratios, the optimal solution requires equal eigenvalues, while the worst-case scenario requires one dominant eigenvalue and the rest approaching zero.This paper investigates the capacity of multi-antenna Gaussian channels in point-to-point communication systems. It considers a system with t transmit antennas and r receive antennas, where the received signal vector v depends on the transmitted vector u through the equation v = Hu + w, with H being the channel matrix and w being Gaussian noise. The capacity of the channel is given by the sum of logarithms of (1 + λ_i P_i / σ²), where λ_i are eigenvalues of HH†, and P_i is the waterfilling power allocation. The paper explores joint optimization of eigenvalues and power allocation to maximize channel capacity. It shows that when the signal-to-noise ratio is low, the optimal solution focuses power on a single channel, while at high signal-to-noise ratios, power is evenly distributed across multiple channels. In the pessimistic capacity analysis, the paper considers the worst-case scenario where an adversary chooses the eigenvalues. It finds that the worst-case capacity occurs when all eigenvalues are equal, while the best-case capacity is achieved with a single nonzero eigenvalue. The paper also analyzes two scenarios: one where the eigenvalues are known only at the receiver, and another where they are known at both ends. In the first case, the optimal power distribution is uniform, while in the second, the minimum achieving eigenvalues can take at most two values. The paper concludes that at low signal-to-noise ratios, the best channel has one nonzero eigenvalue, while the worst channel has all eigenvalues equal. At high signal-to-noise ratios, the optimal solution requires equal eigenvalues, while the worst-case scenario requires one dominant eigenvalue and the rest approaching zero.