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 and/or basis:
“You can’t change what happened between you and your ex-spouse, but you can change your attitude about it. Forgiveness doesn’t mean that what your ex did was right or that you condone what he or she did; it simply means that you no longer want to hold a grudge. Forgiveness is not a gift for the other person; it is a purely selfish act that allows you to put the past behind you.”
—Stephanie Marston (20th century)
“The cultivation of literary pursuits forms the basis of all sciences, and in their perfection consist the reputation and prosperity of kingdoms.”
—Marquês De Pombal (1699–1782)