Jordan matrix

In the mathematical discipline of matrix theory, a Jordan block over a ring R {displaystyle R} (whose identities are the zero 0 and one 1) is a matrix composed of zeroes everywhere except for the diagonal, which is filled with a fixed element λ ∈ R {displaystyle lambda in R} , and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan. In the mathematical discipline of matrix theory, a Jordan block over a ring R {displaystyle R} (whose identities are the zero 0 and one 1) is a matrix composed of zeroes everywhere except for the diagonal, which is filled with a fixed element λ ∈ R {displaystyle lambda in R} , and for the superdiagonal, which is composed of ones. The concept is named after Camille Jordan. Every Jordan block is thus specified by its dimension n and its eigenvalue λ {displaystyle lambda } and is indicated as J λ , n {displaystyle J_{lambda ,n}} .Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix; using either the ⊕ {displaystyle oplus } or the “ d i a g {displaystyle mathrm {diag} } ” symbol, the ( n 1 + … + n r ) × ( n 1 + … + n r ) {displaystyle (n_{1}+ldots +n_{r}) imes (n_{1}+ldots +n_{r})} block diagonal square matrix consisting of r {displaystyle r} diagonal blocks, where the first is J λ 1 , n 1 {displaystyle J_{lambda _{1},n_{1}}} , the second is J λ 2 , n 2 {displaystyle J_{lambda _{2},n_{2}}} , … {displaystyle ldots } , the r {displaystyle r} -th is J λ r , n r {displaystyle J_{lambda _{r},n_{r}}} , can be compactly indicated as J λ 1 , n 1 ⊕ … ⊕ J λ r , n r {displaystyle J_{lambda _{1},n_{1}}oplus ldots oplus J_{lambda _{r},n_{r}}} or d i a g ( J λ 1 , n 1 , … , J λ r , n r ) {displaystyle mathrm {diag} left(J_{lambda _{1},n_{1}},ldots ,J_{lambda _{r},n_{r}} ight)} , respectively.For example the matrix is a 10 × 10 {displaystyle 10 imes 10} Jordan matrix with a 3 × 3 {displaystyle 3 imes 3} block with eigenvalue 0 {displaystyle 0} , two 2 × 2 {displaystyle 2 imes 2} blocks with eigenvalue the imaginary unit i {displaystyle i} , and a 3 × 3 {displaystyle 3 imes 3} block with eigenvalue 7. Its Jordan-block structure can also be written as either J 0 , 3 ⊕ J i , 2 ⊕ J i , 2 ⊕ J 7 , 3 {displaystyle J_{0,3}oplus J_{i,2}oplus J_{i,2}oplus J_{7,3}} or d i a g ( J 0 , 3 , J i , 2 , J i , 2 , J 7 , 3 ) {displaystyle mathrm {diag} left(J_{0,3},J_{i,2},J_{i,2},J_{7,3} ight)} . Any n × n {displaystyle n imes n} square matrix A {displaystyle A} whose elements are in an algebraically closed field K {displaystyle K} is similar to a Jordan matrix J {displaystyle J} , also in M n ( K ) {displaystyle mathbb {M} _{n}(K)} , which is unique up to a permutation of its diagonal blocks themselves. J {displaystyle J} is called the Jordan normal form of A {displaystyle A} and corresponds to a generalization of the diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all 1 × 1 {displaystyle 1 imes 1} . More generally, given a Jordan matrix J = J λ 1 , m 1 ⊕ J λ 2 , m 2 ⊕ … ⊕ J λ N , m N {displaystyle J=J_{lambda _{1},m_{1}}oplus J_{lambda _{2},m_{2}}oplus ldots oplus J_{lambda _{N},m_{N}}} , i.e. whose k th {displaystyle k^{ ext{th}}} diagonal block, 1 ≤ k ≤ N {displaystyle 1leq kleq N} is the Jordan block J λ k , m k {displaystyle J_{lambda _{k},m_{k}}} and whose diagonal elements λ k {displaystyle lambda _{k}} may not all be distinct, the geometric multiplicity of λ ∈ K {displaystyle lambda in K} for the matrix J {displaystyle J} , indicated as g m u l J λ {displaystyle mathrm {gmul} _{J}lambda ,} , corresponds to the number of Jordan blocks whose eigenvalue is λ {displaystyle lambda } . Whereas the index of an eigenvalue λ {displaystyle lambda } for J {displaystyle J} , indicated as i d x J λ {displaystyle mathrm {idx} _{J}lambda ,} , is defined as the dimension of the largest Jordan block associated to that eigenvalue. The same goes for all the matrices A {displaystyle A} similar to J {displaystyle J} , so i d x A λ {displaystyle mathrm {idx} _{A}lambda ,} can be defined accordingly with respect to the Jordan normal form of A {displaystyle A} for any of its eigenvalues λ ∈ s p e c A {displaystyle lambda in mathrm {spec} A} . In this case one can check that the index of λ {displaystyle lambda } for A {displaystyle A} is equal to its multiplicity as a root of the minimal polynomial of A {displaystyle A} (whereas, by definition, its algebraic multiplicity for A {displaystyle A} , m u l A λ {displaystyle mathrm {mul} _{A}lambda ,} , is its multiplicity as a root of the characteristic polynomial of A {displaystyle A} , i.e. det ( A − x I ) ∈ K [ x ] {displaystyle det(A-xI)in K} ).An equivalent necessary and sufficient condition for A {displaystyle A} to be diagonalizable in K {displaystyle K} is that all of its eigenvalues have index equal to 1 {displaystyle 1} , i.e. its minimal polynomial has only simple roots. Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices): the Jordan decomposition is, in general, a computationally challenging task.From the vector space point of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (i.e. via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for. Let A ∈ M n ( C ) {displaystyle Ain mathbb {M} _{n}(mathbb {C} )} (i.e. a n × n {displaystyle n imes n} complex matrix) and C ∈ G L n ( C ) {displaystyle Cin mathrm {GL} _{n}(mathbb {C} )} be the change of basis matrix to the Jordan normal form of A {displaystyle A} , i.e. A = C − 1 J C {displaystyle A=C^{-1}JC} .Now let f ( z ) {displaystyle f(z)} be a holomorphic function on an open set Ω {displaystyle {mathit {Omega }}} such that s p e c A ⊂ Ω ⊆ C {displaystyle mathrm {spec} Asubset {mathit {Omega }}subseteq mathbb {C} } , i.e. the spectrum of the matrix is contained inside the domain of holomorphy of f {displaystyle f} . Let be the power series expansion of f {displaystyle f} around z 0 ∈ Ω ∖ s p e c A {displaystyle z_{0}in {mathit {Omega }}setminus mathrm {spec} A} , which will be hereinafter supposed to be 0 for simplicity's sake. The matrix f ( A ) {displaystyle f(A)} is then defined via the following formal power series