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Jordan Matrix Decomposition

Karol Pąk
2008 Formalized Mathematics  
In this paper I present the Jordan Matrix Decomposition Theorem which states that an arbitrary square matrix M over an algebraically closed field can be decomposed into the form where S is an invertible  ...  matrix and J is a matrix in a Jordan canonical form, i.e. a special type of block diagonal matrix in which each block consists of Jordan blocks (see [13] ).  ...  Then the Jordan block of λ and n is an upper triangular matrix over K of dimension n.  ... 
doi:10.2478/v10037-008-0036-9 fatcat:hol75vbnsbdtrbkw2t6ze2diwa

Outsourcing Computing of Large Matrix Jordan Decomposition

Hongfeng Wu, Jingjing Yan
2019 Mathematical Problems in Engineering  
In this paper, we present a secure, verifiable, efficient, and privacy preserving algorithm for outsourcing Jordan decomposition of large-scale matrix.  ...  However, outsourcing Jordan decomposition of large-scale matrix to the cloud brings great security concerns and challenges since the matrices usually contain sensitive information.  ...  was no outsourcing computing protocol involving matrix Jordan decomposition.  ... 
doi:10.1155/2019/6410626 fatcat:sgodlg3oujal5b55d5cghfog7m

On the Jordan–Chevalley decomposition of a matrix [article]

Meinolf Geck
2022 arXiv   pre-print
The purpose of this note is to advertise an elegant algorithmic proof for the Jordan--Chevalley decomposition of a matrix, following and (slightly) revising the discussion of Couty, Esterle und Zarouf  ...  The Jordan-Chevalley decomposition Let us now fix a matrix A ∈ M n (K). We say that A is nilpotent if A m = 0 for some m ∈ N.  ...  This is called the Jordan-Chevalley decomposition of A. Usually, this is deduced from the Jordan normal form of A (which is a stronger result).  ... 
arXiv:2205.05432v2 fatcat:p6xbscjq3jhk3jbozwdrhxgcii

Orbit decomposition of Jordan matrix algebras of order three under the automorphism groups [article]

Akihiro Nishio, Osami Yasukura
2011 arXiv   pre-print
The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism  ...  , with the trace tr(A) := n i=1 a ii ∈ F, and the adjoint matrix A * := t (ǫA) ∈ M n (K).  ...  Jacobson [16, p.389, Theorem 10]'s minimal polynomial for X ∈ J 3 (K) with respect to the Jordan product.  ... 
arXiv:1007.2926v2 fatcat:hkl6hauwj5eq5fjjlslxss5vc4

DECOMPOSITION OF JORDAN AUTOMORPHISMS OF TRIANGULAR MATRIX ALGEBRA OVER COMMUTATIVE RINGS

XING TAO WANG, YUAN MIN LI
2010 Glasgow Mathematical Journal  
In this paper, we give a complete description of the Jordan automorphisms of T n+1 (R), proving that every Jordan automorphism of T n+1 (R) can be written in a unique way as a product of a graph automorphism  ...  The matrix set {e i,i+k |i = 1, . . . , n − k + 1, k = 1, . . . , n} is a basis of n 1 . Let n 0 = T n+1 (R) and Aut(n k ), k=0,1 denote the Jordan automorphism group of n k , respectively.  ...  Let e ij denote the matrix unit of M n+1 (R) and e the identity matrix of M n+1 (R). The matrix set {e i,i+k |i = 1, . . . , n − k + 1, k = 0, 1, . . . , n} is a basis of T n+1 (R).  ... 
doi:10.1017/s0017089510000406 fatcat:akt4xtzvyfaodl6aosper2yn7m

Decomposition of Jordan automorphisms of strictly triangular matrix algebra over local rings

Xing Tao Wang, Hong You
2004 Linear Algebra and its Applications  
In the cases n = 1, 2, we also give a decomposition for any Jordan automorphism of N n+1 (R) (1 n 2).  ...  In this paper, we prove that any Jordan automorphism of N n+1 (R) can be uniquely written as a product of a graph automorphism, a diagonal automorphism, an inner automorphism and a central automorphism  ...  Let e denote the identity matrix of M n+1 (R) and e ij the matrix with 1 at the position (i, j ) and zeros elsewhere.  ... 
doi:10.1016/j.laa.2004.06.009 fatcat:qtjtiex56rbz7hisphw6vz5x2a

Matrix decompositions over the double numbers [article]

Ran Gutin
2021 arXiv   pre-print
We finish by proposing a new matrix decomposition called the Jordan SVD, which we use to challenge a claim made in Yaglom's book Complex Numbers In Geometry concerning Linear Fractional Transformations  ...  In doing so, we unify some matrix decompositions: For instance, we reduce the LU decomposition of real matrices to LDL decomposition of double matrices; we similarly reduce eigendecomposition of real matrices  ...  decompositions The Jordan SVD generalises the Jordan decomposition.  ... 
arXiv:2105.08047v2 fatcat:fnsibcpwbnbo7jhvct4e4h5fju

A Simple Proof of the Jordan Decomposition Theorem for Matrices

Israel Gohberg, Seymour Goldberg
1996 The American mathematical monthly  
In this paper we use elementary concepts of linear algebra to show that a nilpotent matrix is similar to a Jordan matrix.  ...  Waksman [1] used elementary concepts to show that "-Jordan matrix" is similar to Jordan matrix.  ...  There exists some basis B, in which the matrix representing f in B is a Jordan matrix.  ... 
doi:10.1080/00029890.1996.12004715 fatcat:bitrg44pu5dule7dpxfp37r6za

Spectral Projector-Based Graph Fourier Transforms

Joya A. Deri, Jose M. F. Moura
2017 IEEE Journal on Selected Topics in Signal Processing  
The paper presents the graph Fourier transform (GFT) of a signal in terms of its spectral decomposition over the Jordan subspaces of the graph adjacency matrix A.  ...  Jordan subspaces and the Jordan decomposition are defined in this section.  ...  Consider matrix with distinct eigenvalues 1 , . . . , , ≤ , that has Jordan decomposition = That is, the Fourier transform of , is the unique decomposition = ∑︁ =1 ∑︁ =1̂︀ ,̂︀ ∈ J . (13) The distinct  ... 
doi:10.1109/jstsp.2017.2731599 fatcat:o5lmtypxbjdfnckucpe66vvgwm

Implementation of Complex Matrix Inversion using Gauss-Jordan Elimination Method in Verilog

P. VenkataRao, K.R.K.Sastry K.R.K.Sastry
2015 International Journal of Computer Applications  
It gives the architecture of an optimized complex matrix inversion using GAUSS-JORDAN (GJ) elimination in Verilog with single precision floating-point representation.  ...  Matrix inversion using GJ-elimination improves the frequency when compared with QR Decomposition algorithm. The design is targeted on XC5VLX50T Xilinx FPGA.  ...  The following table shows the required resources for implementing the architecture for matrix inversion using QR decomposition method and Gauss-Jordan elimination method CONCLUSION The implemented architecture  ... 
doi:10.5120/21678-4768 fatcat:tarkc5rhcffixnxrrnys6eevo4

Integral Jordan decomposition of matrices

Inder Bir S. Passi, Klaus W. Roggenkamp, Marcos Soriano
2002 Linear Algebra and its Applications  
We shall relate the existence of a Jordan decomposition for orders to Hochschild cohomology and derive local-global principles for Jordan decomposition.  ...  We treat the question of Jordan decomposition for R-orders, where R is an integrally closed noetherian integral domain with perfect field of quotients K.  ...  In case = R[λ], the order has global integral Jordan decomposition if and only if λ has integral Jordan decomposition.  ... 
doi:10.1016/s0024-3795(02)00355-5 fatcat:7k4hutd7tzaenn7ges572bonoe

A note on the solution of regular PMDs

Liansheng Tan, A. C. Pugh
1999 International Journal of Control  
A new resolvent decomposition of a regular polynomial matrix is derived which is a refinement of the results given by Gohberg et al (1982) and Vardulakis (1991) .  ...  An algorithm for the computation of this resolvent decomposition is provided.  ...  matrix of (q 1 − q j ) × (q 1 − q j ) (j = 2, · · · , k), J i (i = k + 1, · · · , r) are Jordan matrix of (q 1 − 1) × (q 1 − 1).  ... 
doi:10.1080/002071799220227 fatcat:emxpba24zng7lktog4uwodycvu

Designing commutative cascades of multidimensional upsamplers and downsamplers

B.L. Evans
1997 IEEE Signal Processing Letters  
We prove that commutativity is possible if the Jordan canonical form of the rational resampling matrix R = L M , 1 is equivalent to the Smith-McMillan form of R.  ...  The Jordan canonical form decomposes a non-singular matrix R into S J S , 1 .  ...  Theorem 1: Given a square non-singular rational matrix R with Jordan canonical form R =Ŝ Ĵ S , 1 , if J is a diagonal rational matrix, then we can always nd an alternative Jordan canonical form R = S J  ... 
doi:10.1109/97.641397 fatcat:xug3e56eqzb3ljvls42dhxpku4

An analogue of the relationship between SVD and pseudoinverse over double-complex matrices [article]

Ran Gutin
2021 arXiv   pre-print
This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called "Matrix decompositions over the double numbers".  ...  We adopt the idea presented in the same paper that a pair of matrices is actually a single matrix over the double number system.  ...  The Jordan SVD generalises the Jordan decomposition. Let P JP −1 be the Jordan decomposition of a complex matrix A. We then have that [A, A] = U [J, J]V * where U = V = [P, P −1 ].  ... 
arXiv:2105.10455v2 fatcat:zcppda5c3nhrtiex4ecitcgj3m

Decomposition of a monic matrix polynomial into a product of linear factors

I. Krupnik
1992 Linear Algebra and its Applications  
It is proved that a manic matrix polynomial with all its elementary divisors of degree not more than 2 can be decomposed into a product of linear factors. (It is well  ...  Is the decomposition (2) possible in the case when the lengths of all Jordan chains of L(h) are not greater than 2?  ...  -). (2) It is well known that this decomposition in the matrix case is not always possible The following result was achieved in [2-41.  ... 
doi:10.1016/0024-3795(92)90355-e fatcat:qwdh62nrirfobp775q7qlhz65i
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