Why is the spectral norm, the largest singular value? I understand the proof behind this, but I can not intuitively explain why the spectral norm is the largest singular value. It makes sense that it is "The maximum 'scale', by which the matrix can 'stretch' a vector" (Meaning of the spectral norm of a matrix), but I am wondering why the largest singular value is what represents this.
Why is it not a combination of all the singular values? I am also wondering how it relates intuitively and geometrically to the Frobenius norm. Thank you for the clarification.