MATH 311-504Topics in Applied MathematicsLecture 3-2:Complex eigenvalues and eigenvectors.Norm.Fundamental Theorem of AlgebraAny polynomial of degree n ≥ 1, with complexcoefficients, has exactly n roots (counting withmultiplicities).Equivalently, ifp(z) = anzn+ an−1z + ··· + a1z + a0,where ai∈ C and an6= 0, then there exist complexnumbers z1, z2, . . . , znsuch thatp(z) = an(z − z1)(z − z2) . . . (z − zn).Complex eigenvalues/eigenvectorsExample. A =0 −11 0.det(A − λI ) = λ2+ 1 = (λ − i)(λ + i).Characteristic values: λ1= i and λ2= −i.Associated eigenvectors: v1= (1, −i) and v2= (1, i).0 −11 01−i=i1= i1−i,0 −11 01i=−i1= −i1i.v1, v2is a basis of eigenvectors. In which space?ComplexificationInstead of the real vector space R2, we consider acomplex vector space C2(all co mplex numbers areadmissible as scalars).The linear operator f : R2→ R2, f (x) = Ax isreplaced by the complexified linear operatorF : C2→ C2, F (x) = Ax.The vectors v1= (1, −i) and v2= (1, i) form abasis for C2.1 1−i i= 2i 6= 0.Example. Aφ=cos φ −sin φsin φ cos φ.Linear operator L : R2→ R2, L(x) = Aφx is the rotationabout the origin by the angle φ (counterclockwise).Characteristic equation:cos φ − λ −sin φsin φ cos φ − λ= 0.(cos φ − λ )2+ sin2φ = 0.λ1= cos φ + i sin φ = ei φ, λ2= cos φ − i sin φ = e−i φ.Consider vectors v1= (1, −i), v2= (1, i ).cos φ −sin φsin φ cos φ1−i=cos φ + i sin φsin φ − i cos φ= ei φ1−i,cos φ −sin φsin φ cos φ1i=cos φ − i sin φsin φ + i cos φ= e−i φ1i.Thus Aφv1= ei φv1, Aφv2= e−i φv2.Beyond l inear structuren-dimensional coordinate vector space Rncarriesadditional structure: length and dot product.Let x = (x1, x2, . . . , xn), y = (y1, y2, . . . , yn) ∈ Rn.Length: |x| =px21+ x22+ ··· + x2n.Dot product: x · y = x1y1+ x2y2+ ··· + xnyn.Length and dot product =⇒ angle between vectorsAngle: ∠(x, y) = arccosx · y|x||y|.Orthogonality: ∠(x, y) = 90oif x ·y = 0.Properties of the length function:(i) |x| ≥ 0, |x| = 0 only for x = 0 (posi tivity)(ii) |rx| = |r ||x| for all r ∈ R (homogeneity)(iii) |x + y| ≤ |x| + |y| (triangle inequality)Properties of the dot product:(i) x · x ≥ 0, x ·x = 0 only for x = 0 (positivi ty)(ii) x · y = y ·x (symmetry)(iii) (rx ) · y = r(x · y) (homogeneity)(iv) (x + y) · z = x · z + y · z (distributive law)(iii) and (iv) =⇒ x · y is a linear function of x(ii) =⇒ x · y is a linear functio n of y as wellThat is, the dot product is a bilinear functio n.Relation between length and dot product: |x| =√x · xNormThe notion of norm generalizes the no ti on of lengthof a vector in Rn.Definition. Let V be a vector space. A functionα : V → R is called a norm on V if it has thefollowing properties:(i) α(x) ≥ 0, α(x) = 0 onl y for x = 0 (positivity)(ii) α(rx) = |r|α(x ) for all r ∈ R (homogeneity)(iii) α(x + y) ≤ α(x) + α(y) (triangle inequality)Notation. The norm of a vector x ∈ V is usuallydenoted kxk. Different norms on V aredistinguished by subscripts, e.g., kxk1and kxk2.Examples. V = Rn, x = (x1, x2, . . . , xn) ∈ Rn.• kxk∞= max(|x1|, |x2|, . . . , |xn|).Positivity and homogeneity are obvious.The triangle inequality:|xi+ yi| ≤ |xi| + |yi| ≤ maxj|xj| + maxj|yj|=⇒ maxj|xj+ yj| ≤ maxj|xj| + maxj|yj|• kxk1= |x1| + |x2| + ··· + |xn|.Positivity and homogeneity are obvious.The triangle inequality: |xi+ yi| ≤ |xi| + |yi|=⇒Pj|xj+ yj| ≤Pj|xj| +Pj|yj|Examples. V = Rn, x = (x1, x2, . . . , xn) ∈ Rn.• kxkp=|x1|p+ |x2|p+ ··· + |xn|p1/p, p > 0.Theor em k xkpis a norm on Rnfor any p ≥ 1.Remark. kxk2= |x|.Definition. A normed vector space is a vectorspace endowed with a norm.The norm defines a distance function on the normedvector space: dist(x, y) = kx − yk.Then we say that a sequence x1, x2, . . . convergesto a vector x if dist(x, xn) → 0 as n → ∞.Unit circle: kxk = 1kxk = (x21+ x22)1/2blackkxk =12x21+ x221/2greenkxk = | x1| + |x2| bluekxk = max(|x1|, |x2|) redExamples. V = C [a, b], f : [ a, b] → R.• kf k∞= maxa≤x≤b|f (x)| (uniform norm).• kf k1=Zba|f (x)|dx.• kf kp=Zba|f (x)|pdx1/p, p > 0.Theor em k f kpis a norm on C [a, b] for any p ≥
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