Jordan matrix

Jordan matrix:Observation:(1) If 1i n =for each i, then J is a diagonal matrix.(2) If 1i n >for some i, then J is not diagonal(3) If 1i n >for some i, then J is not diagonable.Theorem 3.1.5: Let n A M ∈be a strictly upper triangular matrix. Then there is a nonsingular matrix n S M ∈ and integers 12,,,k n n n with 121k n n n ≥≥≥≥ and 1k i i nn ==∑ such that11(0)00(0)n nk J A S S J -⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦Theorem 3.1.11: Let n A M ∈. Then there is a nonsingular matrix n S M ∈such that1111()00()n nk k J A S S SJS J λλ--⎡⎤⎢⎥==⎢⎥⎢⎥⎣⎦, where 1k i i n n ==∑and 12(){,,,}k A σλλλ= . Furthermore, J is unique up to permutation of the diagonal Jordan blocks.Jordan Canonical Forms: Every matrix n A M ∈is similar to one and only one Jordan matrix.11()0~0()n nk k J A J λλ⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦(1) The number of Jordan blocks in J is the number of linearly independent eigenvectors belonging to J(2) J is diagonableiff k n =(3) The number of Jordan blocks associated with λ(number of Jordan blocks ()l J λ for some l ) is the generic multiplicity of λ; that is, the dimension of the eigenspace corresponding to λ. The sum of the sizes of the Jordan blocks corresponding to λ is the algebraic multiplicity of λ.(4) The size of Jordan blocks in J can be determined by using the function ()()m m r rank J I λλ=- for 1,2,m =Set 0()r n λ=. From the proof of theorem 3.1.11, 1()()m m r r λλ+-is the number of Jordan blocks ()k J λ with size k m ≥.Polynomials and matricesBy the Cayley-Hamilton theorem, n A M ∈ satisfies ()A P λ; that is, ()A P A is the zero matrix.Are there often polynomials that are satisfied by n A M ∈?Observation: If n A M ∈satisfies ()p t , then n A M ∈ satisfies ()cp t for c ∈ .Definition: A polynomial ()p t is called monic if the leading coefficient is 1.Theorem 3.3.1: Let n A M ∈. A satisfies a unique monic polynomial ()A q t of minimum degree. The degree of this monic polynomial is at most n. if ()p t is any polynomial such that ()0p A =, then ()|()A q t p t .Definition: The unique minimum degree monic polynomial satisfied by n A M ∈is called the minimal polynomial of A and denoted ()A q t .Corollary 3.3.3: If n A M ∈and n B M ∈are similar, then ()()A B q t q t =.Corollary 3.3.4: Let n A M ∈. Then ()0A q λ=iff ()A λσ∈.Hence ()A q λ and ()A p λ have the same roots.Let 12(){,,,}m A σλλλ= . 1()()m ri A i i p t t λ==-∏where ri is the multiplicity of i λ. Then 1()()m si A ii q t t λ==-∏ where 1i i s r ≤≤ for each i.Theorem 3.3.6: Let n A M ∈and 12(){,,,}m A σλλλ= . Then 1()()m si A i i q t t λ==-∏ where i s is the size of thelargest Jordan block of A associated with i λ. Hermitian matrix:Definition: n A M ∈is Hermitian if *A A =.n A M ∈is skew-Hermitian if *A A =-Observations:,n A B M ∈(1) *,*,*A A AA A A +areHermitian. (*)***A A A A A A +=+=+(2) If A is Hermitian, then k A is Hermitian.()*(*)k k k A A A ==. Also 1A - is Hermitian.(3) If A and B are Hermitian, then aA bB + is Hermitianfor ,a b ∈.()***aA bB aA bB aA bB +=+=+(4)(*)*A A -is skew-Hermitian. (*)**(*)A A A A A A -=-=--.(5) If A is Hermitian, then iA is skew-Hermitian. ()**iA iA iA ==-Theorem 4.1.2: Each n A M ∈ can be written uniquely as A S iT =+ where S and T are both Hermitian. Theorem 4.1.4: Let n A M ∈. Then A is Hermitian if and only if the following hold.(1) *x Ax is real for x ∈(2) A is normal and all eigenvalue of A are real.(3) *S AS isHermitian for all n S M ∈.Theorem 4.1.5: A is Hermitian if and only if there is a unitary matrix n U M ∈ and a real diagonal matrix n M Λ∈ such that *A U U =Λ. A is real and Hermitianiff there is a real matrix n P M ∈ and a real diagonal matrix n M Λ∈ such that T A P P =Λ.Definition: A Hermitian matrix n A M ∈ is called positive definite if *0x Ax > for all nonzero x ∈ If *0x Ax ≥ then n A M ∈ is called positive semidefinite.n A M ∈isHermitian if and only if *x Ax is real for all x ∈Observation:(1) Any principal submatrix of a positive definite matrix is positive definite.Definition: Let n A M ∈ and let S be a subset subsetof {1,2,,}n . Then ()A S is the matrix obtained from A by deleting the rows and columns corresponding to the elements in S. ()A S -principal submatrix of A.(2) Any nonnegative linear combination of positive semidefinite matrices is positive semidefinite. Let A and B be positive semidefinite and let 0a ≥and 0b ≥. Then*()**(*)(*0x aA bB x x aAx x bBx a x Ax b x Bx +=+=+≥）Hence aA bB + is positive semidefinite.(3) Each eigenvalue of a positive definite matrix is a positive real number.Let ()A λσ∈ and let Ax x λ= for x ∈ . Then **(*)x Ax x x x x λλ== and so **x Ax x xλ=which is positive and real. (4) The trace, determinant and principal minors of a positive definite matrix are positive. Theorem 7.2.1: Let A be a Hermitian matrix.n A M ∈is positive definite if and only if 0λ>for all ()A λσ∈.n A M ∈is positive semidefinite if and only if 0λ≥ for all ()A λσ∈.Theorem 7.2.5: Let n A M ∈be Hermitian. Then A is positive definite if and only if det()0i A >for 1,,i n = . Definition: (1,,)i A A i n =+ for 1,,i n = . 1[]n A a =Theorem 7.3.5 (The SVD theorem):Let ,m n A M ∈. Then A has a singular value decomposition. Observations: *A U V =∑1. The singular values of ,m n A M ∈ are unique since the eigenvalues of *A A are unique. However, U and V are not unique.Singular values of 12{,,}n A σσσ= , 12(*){,,}n A A σλλλ=, i σ=i .2. For 12[,,]n V v v v = ; j v is an eigenvector of *A A belonging to j λ. This is because V diagonalizes *A A .3. We have that j j j Av u σ= for 1,2,,j n = since AV U =∑,1j j j u Av σ=⇒this is used for all the nonzero singular values4. If A has rank r, (i) 12,,r u u u form an orthonormal basis for the column space of A. (ii) 1,r n u u + is an orthonormal basis for (*)N A .5. The rank of A is equal to the number of nonzero singular values of A.Theorem A: Let ,*m n A U V M =∑∈. If X M ∈ such that min F F X M A x A x ∈-=-, then 12221()k n F A x σσ+-=++ In particular, if '''*A U V =∑ where '∑=1200000k σσσ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦then 'X A = *A U V =∑, 111222***n n n A u v u v u v σσσ=+++If A has rank n, then 1211111110'***00n n n n A U V u v u v σσσσσ----⎡⎤⎢⎥⎢⎥⎢⎥==++⎢⎥⎢⎥⎢⎥⎣⎦is the matrix of rank n-1 closest to A w.r.t. the Frobenius norm.If A has rank r, then 111111110'***000r r r r A U V u v u v σσσσ----⎡⎤⎢⎥⎢⎥⎢⎥==++⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦is the matrix of rank r-1 closest to A w.r.t. the Frobenius norm. (0r n <<).Observations:(1) If n A M ∈ is nonsingular, then 'A is singular.'n F A A σ-= (This n σ can be a measure of how close A s to being singular)(2) Find the SVD for 2212-⎡⎤⎢⎥⎣⎦. Find the matrix of rank 1 that is closest to 2212-⎡⎤⎢⎥⎣⎦ w.r.t the Frobenius normLU factorization: If n A M ∈ can be reduced to a strict upper triangular matrix using only row operation 3, thenA can be written as a product of a lower triangular matrix and an upper triangular matrix. QR factorization: If ,m n A M ∈and ()rank A n =, then A can be written as the product QR and ,m n Q M ∈ has orthonormal columns and n R M ∈ is upper triangular with positive diagonal entries. Gram-Schmidt process:Let 1{,,}n x x be a basis for V. Let 1111u x x = and define 1111()k k k k ku x p x p +++=-- for 1,,1k n =- where 11111111,,()()T T k k k k k k k k k p x u u x u u u x u u x u ++++=<>++<>=++Then 1{,,}n u u is an orthonormal basis for V.。