9.4. Cooperativity[[Student version, January 17, 2003]] 313
c. It’s possible that Equation 9.16 will have no real solutions. Show on the contrary that it will
always have two solutions ifbc≥0.
d. Show that furthermore the two eigenvalues will be different (not equal to each other) ifbc >0.Your Turn 9e
Continuing the previous problem, consider asymmetric 2 × 2 matrix, that is, one withM 12 =M 21.
Show that
a. It always has two real eigenvalues.
b. The corresponding eigenvectors are perpendicular to each other, if the two eigenvalues are not
equal.9.3.2 Matrix multiplication
Here is another concrete example. Consider the operation that takes a vectorv,rotates it through
an angleα,and stretches or shrinks its length by a factorg.You can show that this operation
is linear, that its matrix representation isR(α, g)=
[ gcosαgsinα
−gsinαgcosα]
,and that it hasnoreal
eigenvectors (why not?).
Suppose we apply the operationRtwice in a row to a vector.
Your Turn 9f
a. EvaluateM(Nv)for two arbitrary 2× 2 matricesMandN. (That is, apply Equation 9.14
twice.) Show that your answer can be rewritten asQv,whereQis a new matrix called the
productNandM,orsimplyMN.FindQ.
b. Evaluate the matrix productR(α, g)R(β, h), and show that it too can be written as a certain
combination of rotation and scaling. That is, express it asR(γ, c)for someγandc.Findγand
cand explain why your answers make sense.
T 2 Section 9.3.2′on page 341 sketches the generalizations of some of the above results to higher-
dimensional spaces.
9.4 Cooperativity
9.4.1 The transfer matrix technique allows a more accurate treatment of bend cooperativity
Section 9.2.2 gave a provisional analysis of DNA stretching. To begin improving it, let’s make an
inventory of some of the simplifications made so far:
- Wetreated a continuous elastic rod as a chain of perfectly stiff segments, joined by perfectly
free joints. - Wetreated the freely jointed chain as being one-dimensional (Section 9.2.2′on page 340
discusses the three-dimensional case). - Weignored the fact that a real rod cannot pass through itself.