tailieunhanh - Handbook of mathematics for engineers and scienteists part 32

A polygon is said to be plane if its vertices are coplanar. A polygon is said to be simple if its nonadjacent sides do not have common interior or endpoints. A polygon is said to be convex if it lies on one side of any straight line passing through two neighboring vertices (Fig. ). In what follows, we consider only plane simple convex polygons. | . Matrices AND Determinants 185 Example 3. Consider the real symmetric matrix 11 -6 2 A A I -6 10 -4 . A 2 -4 6 Its eigenvalues are A1 18 A2 6 A3 3 and the respective eigenvectors are 2 X1 2 2 X2 1 A 2 2 X3 -2 Consider the matrix S with the columns X1 X2 and X3 1 2 2 S I 2 1 -2 . 2 -2 1 J Taking A1 ST AS we obtain a diagonal matrix 12 2 A 11 -6 2 A 12 2 A 27 0 0 A A1 ST AS I 2 1 -2 -6 10 -4 2 1 -2 I 0 54 0 . A2 -2 1 A 2 -4 6 A2 -2 1 A 0 0 162 Taking A2 S 1 AS we obtain a diagonal matrix with the eigenvalues on the main diagonal 1 -3 -6 -6 A 11 -6 2 A 1 2 2 A 3 0 0 A A2 S AS -1 -6 -3 6 -6 10 -4112 1 -2 I 0 6 0 . 27 A-6 6 -3 J A 2 -4 6 2 -2 1 A0 0 18 We note that A1 9A2. . Characteristic equation of a matrix. The algebraic equation of degree n a A det A AI det av Aỏij ail A a21 a12 ữ22 A a1n a2n an1 an2 ann 0 is called the characteristic equation of the matrix A of size n X n and Ja A is called its characteristic polynomial. The spectrum of the matrix A . the set of all its eigenvalues coincides with the set of all roots of its characteristic equation. The multiplicity of every root Ai of the characteristic equation is equal to the multiplicity mi of the eigenvalue Ai. Example 4. The characteristic equation of the matrix 4-8 1 A A I 5-9 1 A 4 -6 -1 has the form 4- A -8 1 A Ia X det I 5 -9-A 1 -A3-6A2-11A-6 - A 1 A 2 A 3 . A 4 -6 -1 -A Similar matrices have the same characteristic equation. Let Aj be an eigenvalue of a square matrix A. Then 1 aAj is an eigenvalue of the matrix aA for any scalar a 2 Aj is an eigenvalue of the matrix Ap p 0 1 . N for a nondegenerate A otherwise p 0 1 . N where N is a natural number 3 a polynomial f A of the matrix A has the eigenvalue f A . 186 Algebra Suppose that the spectra of matrices A and B consist of eigenvalues Xj and p respectively. Then the spectrum of the Kronecker product A B is the set of all products Xj p . The spectrum of the direct sum of matrices A Ai . An is the union of the spectra of the matrices A1 . .