Definition 1.2 Ae(A)is the set of zEc such that zEA(A+E)for some EE cmxn with IE≤e. Definition1.3Ae(A)is the set of z∈C such that‖(zi-A)vl≤e for some v∈Cn with wl=1. Definition 1.4 (assuming that the norm is.2)A(A)is the set of zEC such that omin(zI-A)≤e, (1.3) where omin denotes the smallest singular value. In section 2 we give a new definition of pseudospectra.In section 3 we consider some fun- damental properties of this new definition.In section4 we present some numerical examples to examine our conclusions.For simplicity,our norm.will always be the vector 2-norm. 2 A new definition of matrices pseudospectra Let B =2-A =[b1,62,...,on].It is shown that a system of vectors (o1,b2,...,on}is de- pendence if and only if G[b1,b2,...,n]=0,where G[61,b2,...,bn]is Gram determinant,i.e., G[b1,b2,...,n]=det(B*B).We can see that if 2A(A)is an eigenvalue of A then we must have deti(B*B)=0.Based on this consideration we give another definition of pseudospectra. On the other hand,let A be an m x n matrix with m>n,we write A as follows, A=[a1,a2,...,an]. (2.1) A system of vectors {a,a2,..,1kn is e-linear dependence,if Ga,a2,..ak for any given e>0[3].Obviously,if a system of vectorsfa1,a2,...,ak his e-linear dependence then a system of vectorsfai,a2,...,ar}with r>k is also c-linear dependence.And we can have the following result[4]. Suppose (61,b2,...,b&}is an orthogonal system and lbill =llaill,i 1,2,...,k then G[a1,a2,,a]≤G[b1,b2,…,bk (2.2) The equality is satisfied if and only if fal,a2,..,ak}is also an orthogonal system. Based on this consideration we give a new definition of pseudospectra. Definition 2.1 Let AE Cmxn and e>0 be arbitrary.The e-pseudospectrum Ac(A)of A is the set of z∈C such that c(A)={z∈C:G2(zi-A)=G[b1,b2,,bnl≤e} (2.3) As we will show,A(A)depends continuously on A(for e>0)and is nonempty for sufficiently large e. 2Definition 1.2 Λ²(A) is the set of z ∈ C such that z ∈ Λ(A + E) for some E ∈ Cm×n with kEk ≤ ². Definition 1.3 Λ²(A) is the set of z ∈ C such that k(z ˜I − A)νk ≤ ² for some ν ∈ Cn with kνk = 1. Definition 1.4 (assuming that the norm is k.k2) Λ²(A) is the set of z ∈ C such that σmin(z ˜I − A) ≤ ², (1.3) where σmin denotes the smallest singular value. In section 2 we give a new definition of pseudospectra. In section 3 we consider some fun￾damental properties of this new definition.In section4 we present some numerical examples to examine our conclusions. For simplicity, our norm k.k will always be the vector 2-norm. 2 A new definition of matrices pseudospectra Let B = z ˜I − A = [b1, b2, ..., bn]. It is shown that a system of vectors {b1, b2, ..., bn} is de￾pendence if and only if G[b1, b2, ..., bn] = 0, where G[b1, b2, ..., bn] is Gram determinant, i.e., G[b1, b2, ..., bn] ≡ det(B∗B). We can see that if z ∈ Λ(A) is an eigenvalue of A then we must have det 1 2 (B∗B) = 0. Based on this consideration we give another definition of pseudospectra. On the other hand , let A be an m × n matrix with m ≥ n, we write A as follows, A = [a1, a2, ..., an]. (2.1) A system of vectors {a1, a2, ..., ak}, 1 ≤ k ≤ n is ²−linear dependence, if G 1 2 [a1, a2, ..., ak] ≤ ² for any given ² ≥ 0[3]. Obviously, if a system of vectors{a1, a2, ..., ak}is ²−linear dependence then a system of vectors{a1, a2, ..., ar} with r > k is also ²−linear dependence. And we can have the following result[4]. Suppose {b1, b2, ..., bk} is an orthogonal system and kbik = kaik, i = 1, 2, ..., k then G[a1, a2, ..., ar] ≤ G[b1, b2, ..., bk] (2.2) . The equality is satisfied if and only if {a1, a2, ..., ak} is also an orthogonal system. Based on this consideration we give a new definition of pseudospectra. Definition 2.1 Let A ∈ Cm×n and ² ≥ 0 be arbitrary.The ²−pseudospectrum Λ²(A) of A is the set of z ∈ C such that Λ²(A) = {z ∈ C : G 1 2 (z ˜I − A) = G 1 2 [b1, b2, ..., bn] ≤ ²} (2.3) As we will show, Λ²(A) depends continuously on A(for ² > 0) and is nonempty for sufficiently large ². 2