The core principle establishes that a linear transformation can be represented by distinct matrices depending on the chosen basis within vector space $\mathbb{R}^n$, where these representations form similar matrices related via change-of-basis transformations. This theoretical framework relies on formal definitions of eigenvector bases, orthonormality, and diagonalization to simplify complex operators into computationally efficient forms for signal representation. Operating at the intersection of linear algebra and digital image processing, this concept provides the mathematical mechanism enabling lossy compression by identifying basis vectors that capture high-energy components (low frequencies) while allowing low-magnitude coefficients (high frequencies or noise) to be discarded without perceptible degradation.
The core principle establishes that a linear transformation can be represented by distinct matrices depending on the chosen basis within vector space $\mathbb{R}^n$, where these representations form …