CHANGE OF BASIS AND LINEAR OPERATORS

JAN MANDEL

Supplement to Lay’s Linear algebra, Sec. 5.4 1. Notation. V is a vector space and = b1, . . . , bn its basis • B { } W is a vector space and = c1, . . . , cm its basis • T : V W is a linear operatorC { } • → 2. Matrix of a linear operator. Consider x V with coordinates [x] = ∈ B (x1, . . . , xn),

x = x1b1 + + xnbn. ··· Applying to this the linear operator T we get

T x = x1T b1 + + xnT bn. ··· Applying the linear operator y [y] , we get 7→ C [T x] = x1[T b1] + + xn[T bn] , C C ··· C hence

x1  .  [T x] = [T b1] ,..., [T bn] . , C  C C    xn  so [T x] = T [x] . C C←B B where

T = [T b1] ,..., [T bn] C←B  C C is the matrix of the operator T relative to the bases and . B C 3. Special cases. 1. If V = W and T = I is the identity operator I : x x on V , the matrix of I relative to the bases and is the change of coordinates7→ matrix from to , B C B C I = [b1] ,..., [bn] ] = P . C←B  C C C←B (P is the notation for the same thing from Sec. 4.7). 2. If V = Rn, W = Rm, and are the canonical bases, that is columns of the identity matrices ofB size nCand m, respectively, and T : x Ax, where 7→ A = [a1, . . . , an] is m n matrix, then × T = [T b1] ,..., [T bn] = [a1, . . . , an] = A. C←B  C C 3. If V = W = Rn, P the matrix having as columns the vectors of , and = n B E e1, . . . , en is the canonical basis in R , then P is the change of coordinates {matrix from} to , B E P = [b1, . . . , bn] = I . E←B 1 4. Change of basis and matrix of a linear operator. Let T : V V and , are two bases of V . Then → B C [T x] = T [x] B B←B B and

[T x] = I [T x] C C←B B = I T [x] C←B B←B B 1 = I T I− [x] , C←B B←B C←B C which gives

1 T = I T I− . C←C C←B B←B C←B 5. Similarity of matrices and diagonalization. Consider n n matrix A, the linear operator T : x Ax in Rn, and a regular matrix P . With× the basis formed by the columns of P7→and the canonical basis (=columns of identityB matrix), we have E

A = T E←E 1 = I T I− E←B B←B 1 E←B = PDP − ,D = T . B←B 1 If two matrices A and D are related by A = PDP − for some matrix P , we say that they are similar. We see that two similar matrices can be always understood to be matrices of the same linear operator relative to two different bases. 1 6. Diagonalization. A particularly important case is when A = PDP − with D a diagonal matrix. Then we say that A is diagonalizable, and we know from Sec. 5.3 that the diagonal entries of D are eigenvalues and the columns of P are egenvectors of A. Finding such D and P is called diagonalization of A, and it is the same as finding a basis in which the matrix of the operator x Ax is diagonal. This is a basis formed of eigenvectors of A. 7→ 1 7. Transformation of eigenvectors. If A = PDP − then the following is equivalent:

Au = λu 1 PDP − u = λu 1 1 DP − u = λP − u.

1 Hence, the eivenvalues of A and PDP − are same, and the eigenvectors u of A and 1 eigenvectors v of D are related by v = P − u, that is, u = P v.

2