MATH 311-504Topics in Applied MathematicsLecture 2-10:Matrix of a linear transformation (continued).Eigenvalues and eigenvectors.Matrix transformationsTheorem Suppose L : Rn→ Rmis a linear map. Thenthere exists an m ×n matrix A such that L(x) = Ax for allx ∈ Rn. Columns of A are vectors L(e1), L(e2), . . . , L(en),where e1, e2, . . . , enis the standard basis for Rn.y = Ax ⇐⇒y1y2...ym=a11a12. . . a1na21a22. . . a2n............am1am2. . . amnx1x2...xn⇐⇒y1y2...ym= x1a11a21...am1+ x2a12a22...am2+ · · · + xna1na2n...amnCoordinatesIf {v1, v2, . . . , vn} is a basis for a vector space V ,then any vector v ∈ V has a unique representationv = x1v1+ x2v2+ · · · + xnvn,where xi∈ R. The coefficients x1, x2, . . . , xnarecalled the coordinates of v with respect to theordered basis v1, v2, . . . , vn.The coordinate mappingvector v 7→ i ts coordinates (x1, x2, . . . , xn)provides a one-to-one correspondence between Vand Rn. Besides, this m apping is l inear.Matrix of a linear transformationLet V , W be vector spaces and f : V → W be a linear map.Let v1, v2, . . . , vnbe a basis for V and g1: V → Rnbe thecoordinate mapping corresponding to this basis.Let w1, w2, . . . , wmbe a basis for W and g2: W → Rmbe the coordinate mapping corresponding to this basis.Vf−→ Wg1yyg2Rn−→ RmThe composition g2◦f ◦g−11is a linear mapping of Rnto Rm.It is represented as x 7→ Ax, where A is an m×n matrix.A is called the matrix of f with respect to bases v1, . . . , vnand w1, . . . , wm. Columns of A are coordinates of vectorsf (v1), . . . , f (vn) with respect to the basis w1, . . . , wm.Example. L : R2→ R2, Lxy=1 10 1xy.The matrix of L with respect to the standard basise1= (1, 0), e2= (0, 1) is1 10 1.The matrix w.r.t. the basi s v1= (3, 1), v2= (2, 1)is2 1−1 0since L(v1) = 2v1− v2, L(v2) = v1.The matrix w.r.t. the bas is w1= (0, 1), w2= (1, 0)is1 01 1since L(w1) = w1+ w2, L(w2) = w2.Eigenvalues and eigenvectorsDefinition. Let V be a vector space and L : V → Vbe a linear operator. A number λ i s called aneigenvalue of the operator L if L(v) = λvfor anonzero vector v ∈ V . The vector v is called aneigenvector of L associ ated with the eigenval ue λ.Remarks. • Alternative notation:eigenvalue = characteristic value,eigenvector = characteristic v ector.• The zero vector is never considered aneigenvector.• If V is a functional space then eig envectors arealso called eigenfunctions.Example. L : R2→ R2, Lxy=2 00 3xy.2 00 310=20= 210,2 00 30−2=0−6= 30−2.Hence (1, 0) is the eigenv ector of L associated withthe eigenvalue 2 while (0, −2) is the eigenvector ofL associated with the eigenvalue 3.Remark. Eigenvalues and eigenvectors of a matrixtransformati on L : Rn→ Rn, L(x) = Ax are alsocalled eigenvalues and eigenvectors of the matrix A.Example. L : R2→ R2, Lxy=0 11 0xy.0 11 011=11,0 11 01−1=−11.Hence (1, 1) is the eigenv ector of L associated withthe eigenvalue 1 while (1, −1) is the eigenvector ofL associated with the eigenvalue −1.Vectors v1= (1, 1) and v2= (1 , −1) form a basisfor R2. The matrix of L with respect to this bas is is1 00 −1since L(v1) = v1, L(v2) = −v2.EigenspacesLet L : V → V be a linear operator.For any λ ∈ R, let Vλdenotes the set of alleigenvectors of L as s ociated w ith the eigenvalue λ.A vector v ∈ V belongs to Vλif v 6= 0 andL(v) = λv. Then (L − λ)v = 0, where L − λdenotes the linear operator v 7→ L(v) − λv.Thus ei genvectors from Vλare nonzero vectors f romthe null-space Null(L − λ).λ ∈ R is an eigenvalue of L if Null(L − λ) 6= {0}.If Null(L − λ) 6= {0} then it is called theeigenspace of L associated with the eigenvalue λ.How to find eigenvalues and eigenvectors?L : Rn→ Rn, L( x) = Ax, where A ∈ Mn,n(R).(L − λ)(x) = (A − λI )x for al l λ ∈ R and x ∈ Rn.λ is an eigenvalue ⇐⇒ the matrix A − λI is notinvertible ⇐⇒ det(A − λI ) = 0Definition. det(A − λI ) = 0 is called thecharacteristic equation of the matrix A.Eigenvalues λ of A are roots of the characteristicequation. Associated eigenvectors of A are nonzerosolutions of the equation (A − λI )x = 0.Example. A =a bc d.det(A − λI ) =a − λ bc d − λ= (a − λ)(d − λ) − bc= λ2− (a + d)λ + (ad − bc).Example. A =a11a12a13a21a22a23a31a32a33.det(A − λI ) =a11− λ a12a13a21a22− λ a23a31a32a33− λ= −λ3+ c1λ2− c2λ + c3,where c1= a11+ a22+ a33(the trace of A),c2=a11a12a21a22+a11a13a31a33+a22a23a32a33,c3= det A.Example. A =2 11 2.Characteristic equation:2 − λ 11 2 − λ= 0.(2 − λ)2− 1 = 0 =⇒ λ1= 1, λ2= 3.(A − I )x = 0 ⇐⇒1 11 1xy=00⇐⇒1 10 0xy=00⇐⇒ x + y = 0.The general solution is (−t, t) = t(−1, 1), t ∈ R.Thus v1= (−1, 1) is an eigenv ector associatedwith the eigenvalue 1. The correspondingeigenspace is the line s panned by v1.(A − 3I )x = 0 ⇐⇒−1 11 −1xy=00⇐⇒1 −10 0xy=00⇐⇒ x − y = 0.The general solution is (t, t) = t(1, 1), t ∈ R.Thus v2= (1, 1) is an eigenv ector associated withthe eigenvalue 3. The corresponding eigenspace isthe line spanned by v2.Summary. A =2 11 2.• The m atrix A has two eigenvalues: 1 and 3.• The ei genspace of A associated with theeigenvalue 1 is the line t(−1 , 1).• The ei genspace of A associated with theeigenvalue 3 is the line t(1 , 1).• Eigenvectors v1= (−1, 1) and v2= (1, 1) ofthe matrix A form an orthogonal basis for R2.• Geo metrically, the mapping x 7→ Ax is a stretchby a factor of 3 away from the line x + y = 0 inthe orthogonal