Change of Basis
Given a linear map whose matrix is A, in the basis B of the space it transforms vectors coordinates as = A. As vectors change with the inverse of B, its inverse transformation is = B.
Substituting this in the first expression
hence
Therefore the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis.
Therefore linear maps are said to be 1-co 1-contra -variant objects, or type (1, 1) tensors.
Read more about this topic: Linear Map
Famous quotes containing the words change of, change and/or basis:
“A sense of blessedness comes from a change of heart, not from more blessings.”
—Mason Cooley (b. 1927)
“All photographs are there to remind us of what we forget. In this—as in other ways—they are the opposite of paintings. Paintings record what the painter remembers. Because each one of us forgets different things, a photo more than a painting may change its meaning according to who is looking at it.”
—John Berger (b. 1926)
“Far from being the basis of the good society, the family, with its narrow privacy and tawdry secrets, is the source of all our discontents.”
—Sir Edmund Leach (20th century)