Math 110 Post-Midterm 2 Content

Jordan Form

-Jordan basis is a basis such that M(T) is block diagonal where each block matrix has a constant on the diagonal and 1's above the diagonal with 0's elsewhere -Every T has a Jordan basis -For V = C every operator has a jordan basis (find a jordan basis for each general eigenspace w.r.t. T-aI and combine them)

Complex Linear Operators (Misc.)

-Suppose T is a linear operator and p is a polynomial. Then null(p(T)) and range(p(T)) are invariant under T. -Suppose V over C and a_1, . . . , a_n are the distinct eigenvalues of T. Then (a) V = G(a_1, T) + . . . + G(a_n, T) direct sum. (b) each G(a_j, T) is invariant under T (c) each (T-a_jI)|g(a_j, T) is nilpotent -If V over C then there exists a basis of V consisting of generalized eigenvectors of T.

permutation

-a perm of a list (1, . . ., n) is a list (m_1, . . ., m_n) that contains each of the numbers 1, . . ., n exactly once -the set of all permutation is perm(n) -the sign of a permutation is1 if the number of pairs of integers (j, k) with 1<=j<k<=n such that j appears after k in the list (m_1, . . ., m_n) is even and -1 if odd -the sign is (-1)^k where k is the number of transpositions from OG order.

characteristic polynomial

-p = (z-a_1)^d_1 . . . where a_1 is the complexified eigenvalue of T and d_1 is its multiplicity -det(zI-T) -degree dimV -p(T) = 0

determinant of linear map

-the determinant of T is the product of the complexified eigenvalues -n = dimV. detT = (-1)^n times the constant term of the characteristic polynomial of T -An operator on V is invertible iff determinant =! 0 -characteristic polynomial = det(zI-T) -determinant of an isometry has absolute value = 1 -|detT| = det(sqrt(T*T))

trace

-the trace of T is the sum of the complexified eigenvalues -if n = dimV, traceT equals the negative of the coefficient of z^(n-1) in the characteristic polynomial of T -trace(A) is the sum of the diagonal entries -trace(AB) = trace(BA) -trace of M(T) is independent of choice of basis -trace(M(T)) = trace(T) -trace(S+T) = trace(S) + trace(T) -there are no operators such that ST-TS = I

Null Spaces of Powers of an Operator

-{0} < null(T^0) < null(T^1) < . . . < null(T^k) < null(T^k+1) < . . . -Suppose m is a nonnegative integer such that null(T^m) = null(T^m+1). Then null(T^m) = null(T^m+1) = null(T^m+2) . . . - Let n = dimV. n is such an m = V = null(T^n) + range(T^n) direct sum

Self-Adjoint

A linear operator T is Self-Adjoint if T* = T. -Every eigenvalue of a self-adjoint operator is real. -If V over C, then <Tv, v> = 0 for all v implies that T = 0. -If V over C, T is self-adjoint iff <Tv, v> is real for all v. -If T is self-adjoint and <Tv, v> = 0 for all v, then T = 0 -If T is self-adjoint and b^2 < 4c, then T^2 + bT + cI is invertible -Self-adjoint operators have an eigenvalue -If T is self-adjoint and U is invariant under T, then (a) U^complement is invariant under T (b) T|u is self-adjoint (c) T|u^complement is self-adjoint

Normal

A linear operator T is normal iff T**T = TT**. -T is normal iff ||Tv|| = ||T*v|| -a eigenvalue -> a conjugate is an eigenvalue -eigenvectors corresponding to distinct eigenvalues of a normal operator are orthagonal

Square Root

An operator R is called a square root of an operator T if R^2 = T -Every positive operator on V has a unique square root -If V = C and T invertible then T has a square root

Isometry

An operator S is an isometry if ||Sv|| = ||v|| for all v. Suppose S is a linear operator. TFAE: (a) S is an isometry (b) <Su, Sv> = <u, v> (c) Se_1, . . . , Se_n is orthanormal for every orthanormal list of vectors e_1, . . . , e_n in V. (d) There exists an orthanormal basis e_1, . . . , e_n of V such that Se_1, . . . , Se_n is orthanormal. (e) S*S = I (f) SS* = I (g) S* is an isometry (h) S is invertible and S^-1 = S* If F over C, then TFAE: (a) S is an isometry (b) There exists an orthanormal basis of V consisting of eigenvectors of S whose corresponding eigenvalues all have absolute value 1.

Positive Operators

An operator T is positive if T is self-adjoint and <Tv, v> >= 0. Suppose T is a linear operator. Then TFAE: (a) T is positive (b) T is self-adjoint and all eigenvalues of T are nonnegative (c) T has a positive square root (d) T has a self-adjoint square root (e) there exists an operator R such that T = R*R

Nilpotent Operators

An operator is called nilpotent if some power of it equals 0. -If N is nilpotent, then N^(dimV) = 0 -suppose N is a nilpotent operator on V. Then there is a basis of V with respect to which the matrix N has the form: | 0 * | | . . . | | 0 0 | here all entries on or below the diagonal are 0. -If N is nilpotent, then N + I has a square root. -N has a basis of the form: N^m_1(v_1) + . . . + v_1, . . . v_n where v_1, . . ., v_n are vectors in V and N^(m_i+1)(v_i) = 0

Block Diagonal Matrix

If V = C and a_i, . . . a_m are distinct eigenvalues of T then there exists a basis of V such that M(T) = |A_1 0 | | . . . | | 0 A_m| Where A_j is an upper triangular matrix with a_j on the diagonal

Change of Basis Formula

Suppose A = M(I, (u_1 . . . ), (v_1, . . . )) -> M(T, (u_1, . . . ,)) = A^-1*M(T(v_1, . . )*A

Adjoint

Suppose T is a linear map from V to W. The adjoint of T is a linear map T** from W to V such that <Tv, w> = <v, T**w> for every v in V and w in W. Properties: (S+T)** = S*** + T** ((a+bi)T)** = (a-bi)T** (T**)** = T I* = I (ST)** = T***S** null(T*) = (rangeT)^complement range(T*) = (nullT)^complement null(T) = (rangeT*)^complement range(T) = (nullT*)^complement

Generalized Eigenvectors/Eigenspaces

Suppose T is a linear operator and a is an eigenvalue of T. A vector v is called a generalized eigenvector of T correponding to a if a =! 0 and (T-aI)^j(v) = 0 for some positive integer j. The generalized eigenspace of T corresponding to a, denoted G(a, T) is the set of all generalized eigenvectors of T corresponding to a and 0. -G(a, T) = null(T-aI)^dimV -Generalized eigenvectors of V corresponding to unique eigenvalues are linearly independent.

Polar Decomposition

Suppose T is a linear operator. Then there exists an isometry S such that T = S(T*T)^(1/2)

Complex Spectral Theorem

Suppose V over C, then TFAE: (a) T is normal (b) V has an orthanormal basis consisting of eigenvectors of T (c) T has a diagonal matrix with respect to some orthanormal basis of V

Real Spectral Theorem

Suppose V over R, then TFAE: (a) T is self-adjoint (b) V has an orthanormal basis consisting of eigenvectors of T (c) T has a diagonal matrix with respect to some orthanormal basis of V

Conjugate Transpose

The conjugate transpose of an m-by-n matrix is the n-by-m matrix obtained by interchanging the rows and columns and then taking the complex conjugate of each entry. -M(T*, (f_1, . . . , f_m), (e_1, . . . , e_n)) is the conjugate transpose of M(T, (e_1, . . . , e_n), (f_1, . . . , f_m)).

Multiplicity

The multiplicity of an eigenvalue a_j of T is dim(G(a_j, T)) = dim(null(T-a_jI)^dimV) -sum of the multiplicities = dimV

Singular Values

The singular values of T are the eigenvalues of (T*T)^(1/2) repeated according to multiplicity.. Suppose T has singular values s_1, . . . , s_n. Then there exists orthanormal bases e_1, . . . , e_n and f_1, . . . , f_n of V such that Tv = s_1<v, e_1>f_1 + . . . + s_n<v, e_n>f_n for all v. The singular values of T are the nonnegative square roots of the eigenvalues of T*T.

Minimal polynomial

The unique monic polynomial of smallest degree such that q(T) = 0 -if p(T) = 0, then p is a multiple of the minimal polynomial -zeroes of the minimal polynomial are precisely the complexified eigenvalues of T

monic polynomial

a polynomial with a leading coefficient of 1.

determinant of a matrix

uniquely defined by these three properties: 1. linearity in row entries 2. two identical rows -> =0 3. I gets mapped to 1 detA = sum on all permutations, p[sign(p)*A_m1, 1 . . . , where (m_1, . . . ,) is p det(AB) = det(BA) = det(A)det(B) -determinant is independent of basis -det(T) = det(M(T))