The matrix 2-norm, also known as the spectral norm, is the maximum singular value of a matrix. It measures the largest possible stretching factor of a matrix on any vector. In other words, it is the square root of the largest eigenvalue of the matrix's conjugate transpose multiplied by itself.
The matrix 2-norm can be calculated using various methods, such as the power iteration algorithm or the singular value decomposition (SVD) method. The SVD method involves decomposing the matrix into three matrices: U, Σ, and V. The matrix 2-norm is then the square root of the largest singular value in the diagonal matrix Σ.
The matrix inf-norm, also known as the maximum absolute row sum norm, is the maximum absolute row sum of a matrix. It measures the largest possible amount of absolute change in a vector due to multiplication by the matrix. In other words, it is the maximum absolute column sum of the absolute values of the matrix.
The matrix inf-norm can be calculated by taking the maximum absolute row sum of the matrix. This can be done by summing the absolute values of each row and then taking the maximum value out of those sums. Alternatively, it can also be calculated using the dual norm, which involves taking the transpose of the matrix and then calculating the matrix 1-norm.
To determine if the matrix 2-norm is greater than the matrix inf-norm, we can compare their values. If the matrix 2-norm is greater, then the maximum stretching factor of the matrix on any vector is larger than the maximum absolute row sum of the matrix. In other words, the matrix has a larger impact on vectors than it does on their absolute values. However, if the matrix inf-norm is greater, then the maximum absolute row sum of the matrix is larger, indicating that the matrix has a larger impact on the absolute values of vectors than it does on the vectors themselves.