week08su - Week 5[reprint week 4 5.4 Eigenvalues and Eigenvectors diagonaliization 1 5.5 Eigenspaces Diagonalization A vector v = 0 in Rn(or in Cn is an

# week08su - Week 5[reprint week 4 5.4 Eigenvalues and...

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Week 5: [reprint, week 4: 5.4 Eigenvalues and Eigenvectors] + diagonaliization 1. 5.5 Eigenspaces, Diagonalization ————— A vector v = 0 in R n (or in C n ) is an eigenvector with eigenvalue λ of an n -by- n matrix A if Av = λv. We re-write the vector equation as ( A - λI n ) v = 0 , which is a homogeneous system with coef matrix ( A - λI ) , and we want λ so that the system has a non-trivial solution. We see that the eigenvalues are the roots of the characteristic polynomial P ( λ ) = 0 , where P ( λ ) = det( A - λI ) . To find the eigenvectors we find the distinct roots λ = λ i , and for each i solve ( A - λ i I ) v = 0 . ————— We collect the terminology and results used for diagonalzation. If P ( λ ) written in factored form is P ( λ ) = ( λ - λ 1 ) m 1 . . . ( λ - λ r ) m r , where the λ j are the distinct roots, we say that λ j has (algebraic) multiplicity m j , or that λ j is a simple root (multiplicity 1) if m j = 1. Let d j = dim( E λ j ) be the dimension of the eigenspace (sometimes called
2 the geometric multiplicity), then the main facts are 1. 1 d j m j ; 2. A is non-diagonalizable exactly when there is some eigenvalue (which must be a repeated root) that is defective, d j < m j ; 3. A is diagonalizable exactly when every eigenvalue is non-defective, d j = m j , for j = 1 , . . . , r ; 4. In particular, if P ( λ ) has distinct roots then A is always diagonalizable (1 d j m j = 1). We recall that m j - the algebraic multiplicity - is the number of times the j th distinct eigenvalue is a root of the characteristic polynomial; and d j - the geometric multiplicity - is the dimension of the λ j th eigenspace (that is, the nullspace of A - λ j I ). We will see that an n -by- n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. First, we have a new definition, matrices A and B are similar if there is an n -by- n matrix P so that P has an inverse and B = P - 1 AP. We also recall that a diagonal matrix D = diag ( d 1 , . . . , d n ) is a square matrix with all entries 0 except (possibly) on the main diagonal, where the entries are d 1 , . . . , d n (in that order).