The Spectral Scale and Linear Pencils

1. The Spectral Scale and Linear Pencils z Christopher M. Pavone University of California, Santa Barbara September 7, 2005 y x

2. Notation:

3. A Self-Adjoint Matrix:

4. A Positive Definite Matrix:

6. Compact: Not Compact:

7. Sets That Are Convex: Sets That Are Not Convex:

8. The Spectral Scale

9. In general….

10. How To Read B(A) for A=A*: • B(A) describes the eigenvalues of A (including multiplicity): -Slopes of line segments in boundary are eigenvalues. -Lengths of line segments describe multiplicity. • B(A) tells us if A ¸ 0, or if A is a projection (i.e., if A2=A): -A ¸ 0 if B(A) contained in 1st quadrant. -A=A2 if B(A) parallelogram with horizontal top. • B(A) describes the numerical range W(A) = {<Ax,x> : ||x||=1} of A: -W(A) = set of all slopes at the origin. • The Spectral Scale of A is symmetric about the point . • Only one boundary (upper or lower) is needed to read the eigenvalue information. • We can also read the trace of A from B(A): -Rightmost point in B(A) is (1,(A)).

11. SELF-ADJOINT EXAMPLES:

14. (NON)SELF-ADJOINT EXAMPLES: = A = A1+iA2 where A1 = A2 = A12 M4(C) random self-adjoint A2 = A12-5A1+I

16. Linear Pencils Spectral problems for linear pencils (and more generally polynomial pencils) arise in different areas of mathematical physics (DE’s, BVP’s, controllable systems, the theory of oscillations and waves, elasticity theory, and hydromechanics).

18. The Main Event

19. The relationship b/w the spectral scale and linear pencils: •  = • Fix A = A1 +iA22 • Fix t =(t1,t2) 2R2 with t12+t22=1. • Let At=t1A1+t2A2. • B(At)={((C),(AtC)) | 0 · C · 1} ½R2. • B(A)={((C),(A1C),(A2C)) | 0 · C · 1} ½R3. • Qt = orthogonal projection of R3 onto span{(1,0,0), (0,t1,t2)}. • t = angle in yz-plane determined by (0,t1,t2) and y-axis. • Rt = rotation about the x-axis through an angle of t.

20. z y x

21. B(A) z y x

23. Recall:

24. How do we spot the real elements of (A1,A2) when looking at B(A)?

27. B(A) z y x

28. Questions? The End