That is, if p A(λ) = λ n + a 1λ n−1 + … + a n I, then p A( A) = A n + a 1 A n−1 + … + a n I = 0. The Cayley–Hamilton theorem states that the characteristic polynomial of A is an annihilating polynomial of A. Invariant SubspacesĪ subspace S of ℂ n is called the invariant subspace or A- invariant if Ax ∈ S for every x ∈ S.Ĭlearly, an eigenvector x of A defines a one-dimensional invariant subspace.Īn A-invariant subspace S ⊆ ℂ n is called a stable invariant subspace if the eigenvectors in S correspond to the eigenvalues of A with negative real parts. If λ 1, λ 2,…, λ n are the n eigenvalues of A, then max |λ i|, i = 1,…, n is called the spectral radius of A. If s = 1, then the eigenvalue is a simple eigenvalue. If an eigenvalue of A is repeated s times, then it is called a multiple eigenvalue of multiplicity s. A nonzero vector y is called a left eigenvector if y * A = λ y * for some λ ∈ ℂ. The vector x is called a right eigenvector (or just an eigenvector) of A. This is equivalent to the following: λ ∈ ℂ is an eigenvalue of A if and only if there exists a nonzero vector x such that Ax = λ x. The zeros of the characteristic polynomial are called the eigenvalues of A. Then the polynomial p A (λ) = det(λ I – A) is called the characteristic polynomial. 2.3.1 The Characteristic Polynomial, the Eigenvalues, and the Eigenvectors of a Matrix In this section, we state some fundamental concepts and results involving the eigenvalues and eigenvectors: rank, range, nulspaces, and the inverse of a matrix. Read more Navigate DownĪ REVIEW OF SOME BASIC CONCEPTS AND RESULTS FROM THEORETICAL LINEAR ALGEBRAīISWA NATH DATTA, in Numerical Methods for Linear Control Systems, 2004 2.3 MATRICES Note that a sufficient condition for the existence of a unique positive semidefinite stabilizing solution of the CARE(DARE) was given Theorem 13.2.6 ( Theorem 13.3.2). To guarantee the existence of such a solution, it will be assumed throughout this section that ( A, B) is stabilizable (discrete-stabilizable) and the Hamiltonian matrix H (symplectic matrix M) does not have an imaginary eigenvalue (an eigenvalue on the unit circle). If this subspace is spanned by ( X 1 X 2 ) and X 1 is invertible, then X = X 2 X 1 − 1 is a stabilizing solution. NUMERICAL SOLUTIONS AND CONDITIONING OF ALGEBRAIC RICCATI EQUATIONSīISWA NATH DATTA, in Numerical Methods for Linear Control Systems, 2004 13.5.1 The Eigenvector and Schur Vector MethodsĪn invariant subspace methods for solving the CARE(DARE) is based on computing a stable invariant subspace of the associated Hamiltonian (symplectic) matrix that is the subspace corresponding to the eigenvalues with the negative real parts (inside the unit circle).
Īny matrix naturally gives rise to two subspaces. Therefore, all of Span a spanning set for V. If u, v are vectors in V and c, d are scalars, then cu, dv are also in V by the third property, so cu + dv is in V by the second property.In other words the line through any nonzero vector in V is also contained in V. If v is a vector in V, then all scalar multiples of v are in V by the third property.Īs a consequence of these properties, we see: Closure under scalar multiplication: If v is in V and c is in R, then cv is also in V.Closure under addition: If u and v are in V, then u + v is also in V.Non-emptiness: The zero vector is in V.Hints and Solutions to Selected ExercisesĬ = C ( x, y ) in R 2 E E x 2 + y 2 = 1 DĪbove we expressed C in set builder notation: in English, it reads “ C is the set of all ordered pairs ( x, y ) in R 2 such that x 2 + y 2 = 1.” DefinitionĪ subspace of R n is a subset V of R n satisfying:.
3 Linear Transformations and Matrix Algebra
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