The outer derivative g′ must be evaluated at the output of the inner function, i.e., at h(x), not at the original input x. This reflects the sequential nature of composition: the change in x affects h, and the sensitivity of g depends on the current value of h.
Conditions: Computing derivative of g(h(x)); Applying chain rule g′(h(x))h′(x)
The outer derivative g′ must be evaluated at the output of the inner function, i.e., at h(x), not at the original input x. This reflects the sequential nature of composition: the change in x affects h, and the sensitivity of g depends on the current value of h.
Conditions: Computing derivative of g(h(x)); Applying chain rule g′(h(x))h′(x)
A matrix is diagonalizable if and only if it possesses a full set of linearly independent eigenvectors (an eigenbasis). The primary benefit is computational efficiency: calculating high powers of the matrix (e.g., A100) becomes trivial because you simply raise the diagonal entries (eigenvalues) to that power, rather than performing repeated matrix multiplication.
Conditions: Matrix has n linearly independent eigenvectors in n-dimensional space; Change of basis matrix P formed by eigenvectors is invertible
A matrix is diagonalizable if and only if it possesses a full set of linearly independent eigenvectors (an eigenbasis). The primary benefit is computational efficiency: calculating high powers of the matrix (e.g., A100) becomes trivial because you simply raise the diagonal entries (eigenvalues) to that power, rather than performing repeated matrix multiplication.
Conditions: Matrix has n linearly independent eigenvectors in n-dimensional space; Change of basis matrix P formed by eigenvectors is invertible
For a selected test pixel, the algorithm compares the corresponding values in the auxiliary layers against user-defined thresholds. If the threshold requirement is met, the cell value is changed to 1, interpreted as built-up growth.
Conditions: A test pixel is selected from the main raster.; Each relevant auxiliary layer value is compared with its user-defined threshold.; Transition occurs only when the threshold requirement is satisfied.
For a selected test pixel, the algorithm compares the corresponding values in the auxiliary layers against user-defined thresholds. If the threshold requirement is met, the cell value is changed to 1, interpreted as built-up growth.
Conditions: A test pixel is selected from the main raster.; Each relevant auxiliary layer value is compared with its user-defined threshold.; Transition occurs only when the threshold requirement is satisfied.