LA 12, 13

LA 12, 13

1. Change of basis and equivalent matrices

   • The matrix P of the identity map IU : U → U using the basis e1, . . . , en in the domain and e'1 , . . . , e'n in the range is called the change of basis matrix from the basis of ei s to the basis of e'i s. Taking P = (σij ), we obtain the above,
     • • In other words, the columns of P are the coordinates of the “old” basis vectors ei with respect to the “new” basis ei' .
         • The change of basis matrix is invertible. More precisely, if P is the change of basis matrix from the basis of eis to the basis of e'i s and Q is the change of basis matrix from the basis of e 0 i s to the basis of eis then P = Q^−1 .
           • let v ∈ U, and let v and v' denote the column vectors associated with v when we use the bases e1, . . . , en and e'1 , . . . , e'n , respectively. Then Pv = v'
             • Let A = (aij ) ibe the m × n matrix of T with respect to the bases {ei} and {fi}, B = (bij ) is the m × n matrix of T with respect to the bases {e'i } and {f'i } of U and V. Let the n × n matrix P = (σij ) be the change of basis matrix from {ei} to {e 0 i }, and let the m × m matrix Q = (τij ) be the change of basis matrix from {fi} to {f 'i }.
               • With the below notation, we have BP = QA, or equivalently B = QAP^−1
                 • Two m×n matrices A and B represent the same linear map from an n-dimensional vector space to an m-dimensional vector space (with respect to different bases) if and only if there exist invertible n × n and m × m matrices P and Q with B = QAP.
                   • Two m × n matrices A and B are said to be equivalent if there exist invertible P and Q with B = QAP
                     • Let A and B be m × n matrices over K. Then the following conditions on A and B are equivalent. (i) A and B are equivalent. (ii) A and B represent the same linear map with respect to different bases. (iii) A and B have the same rank. (iv) B can be obtained from A by application of elementary row and column operations
                       • • Any m × n matrix is equivalent to the matrix Es defined above, where s = rank(A). The form Es is known as a canonical form for m × n matrices under equivalence. This means that it is an easily recognizable representative of its equivalence class
   • Similar matrices
     • In the previous part we studied what happens to the matrix of a linear map T : U → V when we change bases of U and V . Now we look at the case when U = V , where we only have a single vector space V , and a single change of basis. Surprisingly, this turns out to be more complicated than the situation with two different spaces.
       • Theorem 12.4 applies, and with both Q and P replaced by P −1 we find
         • With the notation above, B = P ^−1AP
           • Two n × n matrices over K are said to be similar if there exists an n × n invertible matrix P with B = P ^−1AP

2. Eigenvectors and Eigenvalues

   • A matrix which is similar to a diagonal matrix is said to be diagonalisable.
     • It turns out that the possible entries on the diagonal of a matrix similar to A can be calculated directly from A. They are called eigenvalues of A and depend only on the linear map to which A corresponds, and not on the particular choice of basisLet T : V → V be a linear map, where V is a vector space over K. Suppose that for some non-zero vector v ∈ V and some scalar λ ∈ K, we have T(v) = λv. Then v is called an eigenvector of T, and λ is called the eigenvalue of T corresponding to v. Note that the zero vector is not an eigenvector
       • Suppose we have Av = λv. Then v is called an eigenvector of A, and λ is called the eigenvalue of A corresponding to v.
         • Let A be an n × n matrix. Then λ is an eigenvalue of A if and only if det(A − λIn) = 0
           • For an n×n matrix A, the equation det(A−xIn) = 0 is called the characteristic equation of A, and det(A − xIn) is called the characteristic polynomial of A.
             • Similar matrices have the same characteristic equation and hence the same eigenvalues
               • Suppose that the matrix A is upper triangular. Then the eigenvalues of A are just the diagonal entries aii of A.
                 • Let T : V → V be a linear map. Then the matrix of T is diagonal with respect to some basis of V if and only if V has a basis consisting of eigenvectors of T. Equivalently, let A be an n × n matrix over K. Then A is similar to a diagonal matrix if and only if the space Kn,1 has a basis of eigenvectors of A.
                   • Let λ1, . . . , λr be distinct eigenvalues of T : V → V , and let v1, . . . , vr be corresponding eigenvectors. (So T(vi) = λivi for 1 ≤ i ≤ r.) Then v1, . . . , vr are linearly independent.
                     • If the linear map T : V → V (or equivalently the n × n matrix A) has n distinct eigenvalues, where n = dim(V ), then T (or A) is diagonalisable
                       • Warning! The converse of Corollary 13.14 is not true. If it turns out that there do not exist n distinct eigenvalues, then you cannot conclude from this that the matrix is not diagonalisable
                         • A basis b1, . . . , bn of R n is called orthonormal if (i) bi · bi = 1 for 1 ≤ i ≤ n, and (ii) bi · bj = 0 for 1 ≤ i, j ≤ n and i =/= j.
                           • An n × n matrix A is said to be symmetric if A^T = A
                             • An n × n matrix A is said to be orthogonal if AT = A^−1 or, equivalently, if AA^T = A^TA = In.
                               • An n × n matrix A over R is orthogonal if and only if the rows r1, . . . , rn of A form an orthonormal basis of R n , if and only if the columns c1, . . . , cn of A form an orthonormal basis of R n,1
                                 • Let A be a real symmetric matrix. Then A has an eigenvalue in R, and all complex eigenvalues of A lie in R.
                                   • Let A be a real symmetric matrix, and let λ1, λ2 be two distinct eigenvalues of A, with corresponding eigenvectors v1, v2. Then v1 · v2 = 0
                                     • Let A be a real symmetric n × n matrix. Then there exists a real orthogonal matrix P with P −1AP (= P TAP) diagonal

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