Properties and Computations of Matrix Pseudospectra Yuming Shen2,Jinxi Zhaol and Hongjun Fan3 1.State Key Laboratory for Novel Software Technology,Nanjing University,Nanjing 210093,P.R.China 2.Department of Mathematics,Guangxi Normal University,Guilin 541004.P.R.China 3.Department of Mathematics,Nanjing University,Nanjing 210008,P.R.China Abstract Pseudospectra were introduced as early as 1975 and became popular tool during the 1990s.In this paper,we give a new definition of pseudospectra by using QR decomposition. Some properties of pseduospectra are explored and an algorithm for the computation of pseduospectra is given. Key words and phrases:eigenvalues,pseudospectra,QR decomposition 1 Introduction Let A be an m x n matrix with m >n.An eigenvalue of the matrix A might be defined by the condition (A-λ)w=0, (1.1) for some nonzero n-vector v,where I denotes the m x n'identity'with 1 on the main diagonal and 0 elsewhere.If (A,v)satisfies (1.1),then we have v=0. A2 where A1 denotes the n x n upper part of A.Hence not only (A,v)must be an eigenpair of A1, but v must also be in the nullspace of A2.Obviously,if A is a square matrix,then we get the canonical definition of eigenvalue. Four equivalent definitions of pseudospectra of square matrix were introduced by [1,7,8,9]. Pseudospectra of rectangular matrix has been considered by Toh,Wright and Trefethen[6,11,12] Higham and Tisseur[2].Here we present these equivalent definitions of pseudospectra as follows 12], Definition 1.1 Let A E cmxn and e>0 be arbitrary.The e-pseudospectrum Ae(A)of A is the set of z∈C such that l(zi-A)‖≥e1, (1.2) where'denotes the pseudoinverse and I denotes the mxn identity with 1 on the main diagonal and 0 elsewhere,C denotes the complex plane. 1Properties and Computations of Matrix Pseudospectra Y uming Shen1,2 , J inxi Zhao1 and Hongjun F an1,3 1.State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, P.R.China 2.Department of Mathematics, Guangxi Normal University, Guilin 541004, P.R.China 3.Department of Mathematics, Nanjing University, Nanjing 210008, P.R.China Abstract Pseudospectra were introduced as early as 1975 and became popular tool during the 1990s.In this paper, we give a new definition of pseudospectra by using QR decomposition. Some properties of pseduospectra are explored and an algorithm for the computation of pseduospectra is given. Key words and phrases: eigenvalues, pseudospectra,QR decomposition 1 Introduction Let A be an m × n matrix with m ≥ n. An eigenvalue of the matrix A might be defined by the condition (A − λ˜I)ν = 0, (1.1) for some nonzero n-vector ν, where ˜I denotes the m × n ’identity’ with 1 on the main diagonal and 0 elsewhere. If (λ, ν) satisfies (1.1), then we have à A1 − λIn A2 ! ν = 0, where A1 denotes the n × n upper part of A. Hence not only (λ, ν) must be an eigenpair of A1, but ν must also be in the nullspace of A2. Obviously, if A is a square matrix, then we get the canonical definition of eigenvalue. Four equivalent definitions of pseudospectra of square matrix were introduced by [1, 7, 8, 9]. Pseudospectra of rectangular matrix has been considered by Toh,Wright and Trefethen[6, 11, 12] , Higham and Tisseur[2]. Here we present these equivalent definitions of pseudospectra as follows[12], Definition 1.1 Let A ∈ Cm×n and ² ≥ 0 be arbitrary. The ²−pseudospectrum Λ²(A) of A is the set of z ∈ C such that k(z ˜I − A) † k ≥ ² −1 , (1.2) where ’†’ denotes the pseudoinverse and ˜I denotes the m×n identity with 1 on the main diagonal and 0 elsewhere, C denotes the complex plane. 1