Control Systems Linear Algebra topics L Lanari
outline basic facts about matrices eigenvalues - eigenvectors - characteristic polynomial - algebraic multiplicity eigenvalues invariance under similarity transformation invariance of the eigenspace geometric multiplicity diagonalizable matrix: necessary & sufficient condition diagonalizing similarity transformation a more convenient similarity transformation for complex eigenvalues spectral decomposition not diagonalizable (A defective) case: Jordan blocks Lanari: CS - Linear Algebra
matrices - notations and terminology p x q matrix M = 0 B @ m m m q m m m q C A m p m p,q m pq M = {m ij : i =,,p & j =,,q} transpose M T = m 0 ij : m 0 ij = m ji vector (column) row vector scalar square matrix rectangular matrix v : n v T : n M : p p M : p q (p 6= q) Lanari: CS - Linear Algebra 3
matrices diagonal matrix block diagonal matrix upper triangular matrix 0 B @ 0 B @ upper block triangular matrix m 0 0 0 m 0 0 C 0 A 0 0 m n 0 @ M 0 0 0 M 0 A 0 0 M 3 m m m n 0 m m n 0 0 0 m nn M M 0 M compact notation: M = diag{m i }, compact notation: M = diag{m i }, C A i =,,p i =,,k similarly: lower triangular strictly lower triangular similarly: lower block triangular strictly lower block triangular Lanari: CS - Linear Algebra 4
matrices multiplication by a diagonal matrix 3 3 0 a a n 6 4 7 6 5 4 0 n a n a nn 3 3 a a n 0 6 4 7 6 5 4 a n a nn 0 n 7 5 = 7 5 = 3 a a n 6 4 7 5 a n n a nn n 3 a a n n 6 4 7 5 a n a nn n = diag{ i } A A det M M 0 M =det(m ) det(m ) det M 0 0 M =det(m ) det(m ) special case determinant of a diagonal matrix = product of the diagonal terms special case useful for the eigenvalue computation Lanari: CS - Linear Algebra 5
matrices square matrix non-singular (invertible), det(m) 6= 0 A = A (AB) = B A M = diag{m i } = M = diag{ m i } A A = AA = I A 0 = I det A =det A T det(mn) = det(m)det(n) for square matrices M and N Lanari: CS - Linear Algebra 6
eigenvalues & eigenvectors u A generic transformation Au in general Au and u have different directions u Au particular directions such that Au is parallel to u u Au= u or Au scalar (scaling factor) Lanari: CS - Linear Algebra 7
eigenvalues & eigenvectors eigenvalue i eigenvector ui Aui = i ui (A - i I )ui = 0 ui belongs to the nullspace of (A - ii) for non-trivial u i 6=0solution to exist it has to be det(a - ii) = 0 the eigenvalues i are the solutions of the characteristic polynomial (of order n = dim(a)) p A ( )=det( I A) = n + a n n + a n n + + a + a 0 i solution of p A ( )=det( I A) =0 same solutions of det (A - I) = 0 Lanari: CS - Linear Algebra 8
eigenvalues & eigenvectors p A ( )=det( I A) = n + a n n + a n n + + a + a 0 polynomial with real coefficients set of the n solutions of p A ( ) = 0 is defined as the spectrum of A: ¾(A) being the coefficients real generic eigenvalue i therefore if ( i R then i C ( u i i R i C real components i ( i, u i complex components and i i ) also is a solution pairs u i define ma( i ) = algebraic multiplicity of eigenvalue i as the multiplicity of the solution = i in p A ( ) = 0 Lanari: CS - Linear Algebra 9
eigenvalues & eigenvectors special cases diagonal matrix triangular matrix (upper or lower) 0 m 0 0 0 m 0 B @ 0 C 0 A 0 0 m n 0 m m m n 0 m m n B @ 0 C A 0 0 m nn eigenvalues = {mi} eigenvalues = elements on the main diagonal eigenvalues = {mii} Lanari: CS - Linear Algebra 0
eigenvalues - invariance similarity transformation A T T AT det(t ) 6= 0 same eigenvalues as A eigenvalues are invariant under similarity transformations (proof) def Au i = i u i! u i v T i A = v T i i = i v T i! v T i right eigenvector (column) left eigenvector (row) property v T i u j = ij with ( =0 if i 6= j ± Kronecker ij = if i = j delta Lanari: CS - Linear Algebra
eigenspace the eigenspace V i corresponding to the eigenvalue i of A is the vector space V i = {u R n Au = i u} or equivalently V i = Ker(A - ii) reminder: span{v, v,, v k } = vector space generated by all possible linear combinations of the vectors v, v,, v k Ker(M) (kernel or nullspace of M) is the subspace of all vectors st Mv = 0 basis of Ker(M) is a set of linearly independent vectors which spans the whole subspace Ker(M) consequence: eigenvector u i associated to i is not unique example A( u i )= Au i = i u i = i ( u i ) all belong to the same linear subspace Lanari: CS - Linear Algebra
geometric multiplicity if i eigenvalue with ma( i ) > it will have one or more linearly independent eigenvectors ui 0 0 ex A =, u 0 =, u 0 = A =, with 6= 0, only u 0 = 0 linearly independent linearly independent note that A and A have the same eigenvalues both with ma( ) = same characteristic polynomial p A ( )=( ) Def geometric multiplicity of i as the dimension of the eigenspace associated to i dim(v i )=mg( i ) mg( i )=dim[ker(a ii)] = n rank(a ii) Lanari: CS - Linear Algebra 3
geometric multiplicity useful property ex A = A = 0, (A 0 I)=, (A 0 I)= apple mg( i ) apple ma( i ) apple n 0 0 0 0 0 0 0 mg( )==ma( ) mg( )=< ma( ) ex 0 05 0 05 P = @ 0 0 A projection matrix 05 0 05 0 0 = = u = @ A u = ma( )==mg( ) 0 0 3 = 0 u 3 = @ 0 A 0 @ 0A V V 3 u3 u generic point 45 u Lanari: CS - Linear Algebra 4
diagonalization Def An (n x n) matrix A is said to be diagonalizable if there exists an invertible (n x n) matrix T such that TAT - is a diagonal matrix Th An (n x n) matrix A is diagonalizable if and only if it has n linearly independent eigenvectors since the eigenvalues are invariant under similarity transformations if A diagonalizable T AT = = diag{ i },i=,,n the elements on the diagonal of are the eigenvalues of A Lanari: CS - Linear Algebra 5
diagonalization we need to find T i ui n linearly independent (by hyp) U = u u u n non-singular Au i = i u i, i =,,n in matrix form A u u u n = u u u n 0 B @ 0 0 0 0 C A 0 0 n AU = U Lanari: CS - Linear Algebra 6
diagonalization AU = U being U non-singular, we can define T such that T = U AT = T! A = T T! = T AT therefore the diagonalizing similarity transformation is T st T = U = u u u n alternative Nec & Suff condition for diagonalizability A: (n x n) is diagonalizable if and only if mg( i) = ma( i) for every i distinct eigenvalues (real and/or complex) =) (= A diagonalizable Lanari: CS - Linear Algebra 7
diagonalization note that since distinct eigenvalues (real and/or complex) =) (= distinct eigenvalues (real and/or complex) =) A diagonalizable linearly independent eigenvectors i 0 0 i diagonal but not distinct eigenvalues Lanari: CS - Linear Algebra 8
diagonalization: complex eigenvalues case complex eigenvalues? ( i, i ) i = i + j i u i = u ai + ju bi eigenvalue eigenvector choices diagonalization T = u i u i! D i = T AT = apple i 0 0 i complex elements or real block x (not diagonalization) T = u ai u bi! M i = T AT = apple i i i i real elements real system representation for complex eigenvalues Lanari: CS - Linear Algebra 9
diagonalization simultaneous presence of real and complex eigenvalues If A diagonalizable, there exists a non-singular matrix R such that RAR = diag { r,m r+,m r+3,,m q } with r = diag{,, r} real eigenvalues M i = apple i! i! i i for each complex pair of eigenvalues Lanari: CS - Linear Algebra 0
diagonalization example 4 6 5 4 3 = A = 5 6 65 /3 = ± 0j 0 0 9 3 = u = 45 0 3 05+0j = +0j u = 4 05+j5 u a = 4 3 05 055 u b = 3 0 45 0 T = u u a u b = 05 3 0 4 05 5 0 0 TAT = 4 0 3 0 0 0 5 0 0 Lanari: CS - Linear Algebra
A diagonalizable: spectral decomposition Hyp: A diagonalizable A = U U columns (linearly independent) U = u u u n U = 6 4 v T v T 3 7 5 rows U U = I v T i u j = ij, i,j =,,n v T n A = U U = u u nu n v T v T 6 4 v T n 3 7 5 spectral form of A A = nx i= i u i v T i or eigen-decomposition Lanari: CS - Linear Algebra
A diagonalizable: spectral decomposition A = nx i u i vi T i= column row (n x n) P i = u i v T i is the projection matrix on the invariant subspace generated by ui A = 0 u = 0 u = P v = = v T = v T = 0 0 P = P 0 0 = 0 P v = u P v = u v P v v = u u Lanari: CS - Linear Algebra 3
not diagonalizable case since in general apple mg( i ) apple ma( i ) apple n if A not diagonalizable then mg( i ) < ma( i ) in this case A is said to be defective each i will have mg( i) blocks Jk with k =,, mg( i ), each of dimension nk Jordan block of dimension nk J k = 6 4 i 0 0 0 i 0 0 0 i 0 0 i 3 7 5 the knowledge of this dimension is out of scope R n k n k J k i I Null space has dimension Lanari: CS - Linear Algebra 4
not diagonalizable case (A defective) generalized eigenvector chain of nk generalized eigenvectors (not part of this course) Jordan canonical form (block diagonal) example unique eigenvalue i of matrix A (n x n) mg( i )=p px ma( i )=n = 9 T : k= n k T AT = J = 6 4J 3 7 5 J p with J k R n k n k Jordan block of dim nk Lanari: CS - Linear Algebra 5
special cases if ma( i) = then mg( i) = if mg( i) = then only one Jordan block of dimension ma( i) apple 0 = ma( )= mg( )= if i unique eigenvalue of matrix A and if rank( ii - A) = ma( i) - then only one Jordan block of dimension ma( i) consequence of the rank-nullity theorem applied to A - ii A ii : R n! R n dim (R n )=dim(ker(a ii)) + dim (Im(A ii)) n nullity(a - ii) rank(a - ii) Lanari: CS - Linear Algebra 6
summary A diagonalizable mg( i) = ma( i) for all i A not diagonalizable mg( i) < ma( i) 9 T st T AT = = diag{ i } nx A = i u i vi T i= 9 T st T AT = diag{j k } block diagonal J k = for real & complex i r = diag{,, r} alternative choice for complex ( i, i*) apple i M i = i 6 4 i spectral form mg( i ) Jordan blocks of the form i 0 0 i 0 0 i i 3 7 5 Rn k n k Lanari: CS - Linear Algebra 7
vocabulary English eigenvalue/eigenvector characteristic polynomial algebraic/geometric multiplicity similar matrix spectral form Italiano autovalore/autovettore polinomio caratteristico molteplicità algebrica/geometrica matrice simile forma spettrale Lanari: CS - Time response 8