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:
“—No, no thou hast not felt the lapse of hours!
For what wears out the life of mortal men?
‘Tis that from change to change their being rolls;
‘Tis that repeated shocks, again, again,
Exhaust the energy of strongest souls
And numb the elastic powers.”
—Matthew Arnold (1822–1888)
“The basis of optimism is sheer terror.”
—Oscar Wilde (1854–1900)