MATH 304Linear algebraLecture 36:Complex eigenvalues and eigenvectors.Symmetric and orthogonal matrices.Complex numbersC: complex numbers.Complex number: z = x + iy ,where x, y ∈ R and i2= −1.i =√−1: imaginary unitAlternative notation: z = x + yi.x = real part of z,iy = imaginary part of zy = 0 =⇒ z = x (real number)x = 0 =⇒ z = iy (purely imaginary number)We add and m ultiply complex numbers aspolynomials in i (but keep in mind that i2= −1).If z1= x1+ i y1and z2= x2+ i y2thenz1+ z2= (x1+ x2) + i(y1+ y2),z1z2= (x1x2− y1y2) + i(x1y2+ x2y1).Examples. • (1 + i ) −(3 + 5i) = (1 −3) + (i −5i )= −2 − 4i;• (1 + i )(3 + 5i ) = 1 · 3 + i · 3 + 1 · 5i + i · 5i= 3 + 3i + 5i + 5i2= 3 + 3i + 5i − 5 = −2 + 8i;• (2 + 3i )(2 − 3i) = 4 − 9i2= 4 + 9 = 13;• i3= −i, i4= 1, i5= i.Geometric representationAny complex num ber z = x + iy is represented bythe vector/point (x, y) ∈ R2.yx0rφ0x = r cos φ, y = r sin φ=⇒ z = r(cos φ + i sin φ) = reiφ.Given z = x + iy , the compl ex conjugate of z is¯z = x − iy . The co nj ugacy z 7→ ¯z is the refl ecti onof C in the real line.z¯z = (x + i y )(x −iy ) = x2−(iy)2= x2+ y2= |z|2.z−1=¯z|z|2, (x + iy )−1=x − iyx2+ y2.Fundamental Theorem of AlgebraAny polynomial of degree n ≥ 1, with complexcoefficients, has exactly n roots (counting wi thmultiplicities).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.Characteristic roots: 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 basi s of eigenvectors. In which space?ComplexificationInstead of the real v ector space R2, we consider acomplex vector space C2(all complex 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.Normal matri cesDefinition. An n×n m atrix A is called• symmetric if AT= A;• orthogonal if AAT= ATA = I , i.e., AT= A−1;• normal if AAT= ATA.Theor em Let A be an n×n matrix with realentries. Then(a) A is normal ⇐⇒ there exists an orthonormalbasis for Cnconsisting of eigenvectors of A;(b) A i s symmetric ⇐⇒ there exi s ts an orthonormalbasis for Rnconsisting of eigenvectors of A.Example. A =1 0 10 1 01 0 1.• A is symmetric.• A has three eigenvalues: 0, 1, and 2.• Associated eigenvectors are v1= (−1, 0, 1),v2= (0, 1, 0), and v3= (1, 0, 1), respecti vely.• Vectors1√2v1, v2,1√2v3form an orthonormalbasis for R3.Theor em Suppose A is a normal m atrix. Then forany x ∈ Cnand λ ∈ C one hasAx = λx ⇐⇒ ATx =λx.Thus any normal matrix A shares wi th ATall realeigenvalues and the corresponding eigenvectors.Also, Ax = λx ⇐⇒ Ax = λ x for any matrix Awith real entries.Coro llary All eigenvalues λ of a symm etric matri xare real (λ = λ). All eigenvalues λ of anorthogonal matrix satisfy λ = λ−1⇐⇒ |λ| = 1.Example. Aφ=cos φ −si n φsin φ cos φ.• AφAψ= Aφ+ψ• A−1φ= A−φ= ATφ• Aφis orthogonal• det(Aφ− λI ) = (cos φ − λ)2+ sin2φ.• Eigenvalues: λ1= cos φ + i sin φ = eiφ,λ2= cos φ − i sin φ = e−iφ.• Associated eigenvectors: v1= (1, −i),v2= (1, i).• Vectors v1and v2form a basis for C2.Consider a linear operator L : Rn→ Rn, L(x ) = Ax,where A is an n×n orthogonal matrix.Theor em There exists an orthonormal basis for Rnsuch that the matrix of L relative to this basis has adiagonal block structureD±1O . . . OO R1. . . O............O O . . . Rk,where D±1is a diagonal matrix whose diagonalentries are equal to 1 or −1, andRj=cos φj−si n φjsin φjcos φj, φj∈
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