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:
“Change of fashion is the tax levied by the industry of the poor on the vanity of the rich.”
—Sébastien-Roch Nicolas De Chamfort (1741–1794)
“Lord Angelo is precise,
Stands at a guard with envy, scarce confesses
That his blood flows, or that his appetite
Is more to bread than stone. Hence shall we see
If power change purpose, what our seemers be.”
—William Shakespeare (1564–1616)
““The Love that dare not speak its name” in this century is such a great affection of an elder for a younger man as there was between David and Jonathan, such as Plato made the very basis of his philosophy, and such as you find in the sonnets of Michelangelo and Shakespeare. It is that deep, spiritual affection that is as pure as it is perfect.... It is in this century misunderstood ... and on account of it I am placed where I am now.”
—Oscar Wilde (1854–1900)