22 Chapter 22 Linear Preserver Problems

22
Linear Preserver
Problems
Peter Šemrl
University of Ljubljana
22.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-1
22.2 Standard Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-2
22.3 Standard Linear Preserver Problems . . . . . . . . . . . . . . . . . . 22-4
22.4 Additive, Multiplicative, and Nonlinear Preservers . . . . . 22-7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-8
Linear preservers are linear maps on linear spaces of matrices that leave certain subsets, properties, relations,
functions, etc., invariant. Linear preserver problems ask what is the general form of such maps. Describing
the structure of such maps often gives a deeper understanding of the matrix sets, functions, or relations
under the consideration. Some of the linear preserver problems are motivated by applications (system
theory, quantum mechanics, etc.).
22.1
Basic Concepts
Definitions:
Let V be a linear subspace of F m×n . Let f be a (scalar-valued, vector-valued, or set-valued) function on
V, M a subset of V, and ∼ a relation defined on V.
A linear map φ : V → V is called a linear preserver of function f if f (φ(A)) = f (A) for every A ∈ V.
A linear map φ : V → V preserves M if φ(M) ⊆ M.
The map φ strongly preserves M if φ(M) = M.
A linear map φ : V → V preserves the relation ∼ if φ(A) ∼ φ(B) whenever A ∼ B, A, B ∈ V.
If for every pair A, B ∈ V we have φ(A) ∼ φ(B) if and only if A ∼ B, then φ strongly preserves the
relation ∼.
Facts:
1. If linear maps φ : V → V and ψ : V → V both preserve M, then φψ : V → V preserves M.
Consequently, the set of all linear transformations on V preserving M is a multiplicative semigroup.
2. The set of all bijective linear transformations strongly preserving M is a multiplicative
group.
3. [BLL92, p. 41] Let M ⊆ V be an algebraic subset and φ : V → V a bijective linear map satisfying
φ(M) ⊆ M. Then φ strongly preserves M.
22-1
22-2
Handbook of Linear Algebra
Examples:
1. Let R ∈ F m×m and S ∈ F n×n . The linear map φ : F m×n → F m×n defined by φ(A) = R AS,
A ∈ F m×n , preserves the set of all matrices of rank at most one. In general, such a map does not
preserve this set strongly.
2. If R and S in the previous example are invertible, then φ is a bijective linear map strongly
preserving matrices of rank one. In fact, such a map φ strongly preserves matrices of rank k,
k = 1, . . . , min{m, n}.
3. If m = n and R, S ∈ F n×n , then the linear map A → R AT S, A ∈ F n×n , preserves matrices of
rank at most one. If both R and S are invertible, then φ strongly preserves the set of all matrices of
rank k, 1 ≤ k ≤ n.
4. Assume that m = n. Let R ∈ F n×n be an invertible matrix, c a nonzero scalar, and f : F n×n → F
a linear functional. Then both maps A → c R AR −1 + f (A)I , A ∈ F n×n , and A → c R AT R −1 +
f (A)I , A ∈ F n×n , strongly preserve commutativity; that is, if φ : F n×n → F n×n is any of these two
maps, then for every pair A, B ∈ F n×n we have φ(A)φ(B) = φ(B)φ(A) if and only if AB = B A.
5. Let · be any norm on Cm×n . A linear map φ : Cm×n → Cm×n is called an isometry if φ(A) =
A, A ∈ Cm×n . Thus, isometries are linear preservers of norm functions.
6. Let 1 ≤ k < min{m, n}. A matrix A ∈ F m×n is of rank at most k if the determinant of every
(k + 1) × (k + 1) submatrix of A is zero. Thus, the set of all m × n matrices of rank at most k is an
algebraic subset of F m×n .
7. The set of all nilpotent n × n matrices is an algebraic subset of F n×n . More generally, given a
polynomial p ∈ F [X], the set of all matrices A ∈ F n×n satisfying p(A) = 0 is an algebraic set.
Hence, bijective linear maps on F n×n preserving idempotent matrices, nilpotents, involutions, etc.
preserve these sets strongly.
22.2
Standard Forms
Definitions:
Let V be a linear subspace of F m×n and let P be a preserving property which makes sense for linear maps
acting on V (P may be the property of preserving a certain subset of V or the property of preserving a
certain relation on V or the property of preserving a certain function defined on V). A linear preserver
problem corresponding to the property P is the problem of characterizing all linear (bijective) maps on
V satisfying this property. Very often, linear preservers have the standard forms (see the next section).
Occasionally, there are interesting exceptional cases especially in low dimensions (see later examples).
Let R ∈ F m×m and S ∈ F n×n be invertible matrices. A map φ : F m×n → F m×n is called an
(R, S)-standard map if either φ(A) = R AS, A ∈ F m×n , or m = n and φ(A) = R AT S, A ∈ F n×n .
In many cases we assume that R and S satisfy some additional assumptions.
Let R ∈ F n×n be an invertible matrix, c a nonzero scalar, and f : F n×n → F a linear functional. A map
φ : F n×n → F n×n is called an (R, c , f )-standard map if either φ(A) = c R AR −1 + f (A)I , A ∈ F n×n ,
or φ(A) = c R AT R −1 + f (A)I , A ∈ F n×n .
When we consider linear preservers on proper subspaces V ⊂ F m×n , we usually have to modify the
notion of standard maps. Let us consider the case when V = Tn ⊂ F n×n is the subalgebra of all upper
triangular matrices. The flip map A → A f , A ∈ Tn , is defined as the transposition over the antidiagonal;
that is, A f = G AT G , where G = E 1n + E 2,n−1 + . . . + E n1 (see Example 1). Standard maps on Tn are
maps of the form
A → R AS,
A → R A S,
f
A ∈ Tn ,
A ∈ Tn ,
A → c R AR −1 + f (A)I,
A ∈ Tn ,
22-3
Linear Preserver Problems
and
A → c R A f R −1 + f (A)I,
A ∈ Tn ,
where R and S are invertible upper triangular matrices, c is a nonzero scalar, and f is a linear functional
on Tn .
Facts:
1. Every (R, S)-standard map is bijective. It strongly preserves matrices of rank k, k = 1, . . . , min{m, n}.
Every (R, c , f )-standard map is either bijective, or its kernel is the one-dimensional subspace consisting of all scalar matrices.
2. Let U ∈ Cm×m and V ∈ Cn×n be unitary matrices. Then a (U, V )-standard map preserves singular
values and, hence, all functions of singular values including unitarily invariant norms.
3. Let R ∈ Cn×n be an invertible matrix. Then an (R, R −1 )-standard map on Cn×n preserves spectrum,
idempotents, nilpotents, similarity, zero products, etc.
4. If U ∈ Cn×n is a unitary matrix, then a (U, U ∗ )-standard map defined on Cn×n strongly preserves
the set of all orthogonal idempotents, the set of all normal matrices, numerical range, etc.
5. If A, B ∈ Tn , then (AB) f = B f A f .
Examples:
1.
⎡
a
⎢
⎢0
⎢
⎢0
⎣
0
⎤f
⎡
b
c
d
e
f
g⎥
0
h
⎥
⎢
⎢0
⎥ =⎢
⎥
⎢0
i⎦
⎣
0
0
j
2. A map φ : C2×2 → C2×2 given by
φ
j
0
a
b
c
d
g
d
h
f
c⎥
0
e
⎥
⎥.
b⎥
⎦
0
0
a
−b
d
a
c
=
⎤
i
is a bijective linear map that strongly preserves commutativity but is not of a standard form.
More generally, any bijective linear map φ : C2×2 → C2×2 satisfying φ(I ) = I strongly preserves
commutativity [Kun99]. In higher dimensions there are no nonstandard bijective linear maps
preserving commutativity [BLL92, p. 76].
3. Let W ⊂ Cn×n be any linear subspace of matrices with the property that AB = BA for every pair
A, B ∈ W. Assume that φ : Cn×n → Cn×n is a linear map whose range is contained in W. Then φ
is a nonstandard map preserving commutativity. If n > 1, then it does not preserve commutativity
strongly. A map φ : C2×2 → C2×2 defined by
a
c
φ
b
d
=
a
b
−b
a
a
b
c
is a concrete example of such map.
4. A map φ : R4×4 → R4×4 given by
⎛⎡
a
⎜⎢
⎜ ⎢∗
⎢
φ⎜
⎜⎢
⎝ ⎣∗
∗
b
c
d
⎤⎞
⎡
d
⎤
a
−d
∗ ∗
⎥⎟ ⎢
⎢ −b
⎥⎟ = ⎢
⎥⎟ ⎢ −c
∗ ⎦⎠ ⎣
d
a
⎥
⎥
−b ⎥
⎦
∗ ∗
∗
−d
−c
b
a
∗ ∗
∗ ⎥⎟
c⎥
22-4
Handbook of Linear Algebra
preserves the real orthogonal group, that is, φ(O) is an orthogonal matrix for every orthogonal
matrix O ∈ R4×4 . Note that the right-hand side of the above equation is the standard matrix
representation of the quaternion a +bi +c j +dk. Similar constructions with matrix representations
of complex and Cayley numbers give nonstandard linear preservers of real orthogonal group on
R2×2 and R8×8 . If φ : Rn×n → Rn×n is a linear preserver of orthogonal group and n ∈ {2, 4, 8},
then φ is a (U, V )-standard map with U and V being orthogonal matrices [LP01, p. 601].
5. A linear map φ acting on 4 × 4 upper triangular matrices given by
⎡
a
⎢
⎢0
⎢
⎢0
⎣
0
⎤
b
c
d
e
f
g⎥
0
0
⎡
e
f
g
b
⎤
j
i
h
⎥
⎢
⎢0
⎥ → ⎢
⎥
⎢0
i⎦
⎣
0
h
⎥
⎥
d⎥
⎦
0
j
0
0
a
0
c⎥
is a nonstandard bijective linear map strongly preserving the set of invertible matrices. All that is
important in this example is that A and φ(A) have the same diagonal entries up to a permutation.
22.3
Standard Linear Preserver Problems
Definitions:
Linear preserver problems ask what is the general form of (bijective) linear maps on matrix spaces having
a certain preserving property. When the general form of linear preservers under the consideration is one
of the standard forms, we speak of a standard linear preserver problem. The following list of some most
important standard linear preserver results is far from being complete. Many more results can be found
in the survey [BLL92].
Facts:
1. [BLL92, Theorem 2.2], [LT92, Prop. 3] Let m, n be positive integers and φ : Cm×n → Cm×n a linear
map. Assume that one of the following two conditions is satisfied:
r Let k be a positive integer, k ≤ min{m, n}, and assume that rank φ(A) = k whenever rank A = k.
r φ is invertible and rank φ(A) = rank φ(B) whenever rank A = rank B.
Then φ is an (R, S)-standard map for some invertible matrices R ∈ Cm×m and S ∈ Cn×n .
2. [BLL92, p. 9] Let F be a field with more than three elements, k a positive integer, k ≤ min{m, n},
and φ : F m×n → F m×n an invertible linear map such that rank φ(A) = k whenever rank A = k.
Then φ is an (R, S)-standard map for some invertible matrices R ∈ F m×m and S ∈ F n×n .
3. [BLL92, Theorem 2.6] Let φ : F m×n → F p×q be a linear map such that rank φ(A) ≤ 1 whenever
rank A = 1. Then either
(a) φ(A) = R AS for some R ∈ F p×m and some S ∈ F n×q or
(b) φ(A) = R AT S for some R ∈ F p×n and some S ∈ F m×q or
(c) The range of φ is contained in the set of all matrices of rank at most one.
4. [BLL92, p. 10] Let F be an infinite field with characteristic = 2, Sn the space of all n × n symmetric
matrices over F , and let k be an integer, 1 ≤ k ≤ n. If φ : Sn → Sn is an invertible linear rank k
preserver, then φ(A) = c R AR T for every A ∈ S. Here, c is a nonzero scalar and R an invertible
n × n matrix.
5. [BLL92, Theorem 2.9] Let F be a field with characteristic = 2. Assume that F has more than three
elements. Let φ : Sn → Sm be a linear map such that rank φ(A) ≤ 1 whenever rank A = 1. Then
either
Linear Preserver Problems
22-5
(a) There exist an m × n matrix R and a scalar c such that φ(A) = c R AR T or
(b) The range of φ is contained in a linear span of some rank one m × m symmetric matrix.
6. [BLL92, Theorem 2.7, Theorem 2.8, p. 25] Let φ be a real linear map acting on the real linear space
Hn of all n × n Hermitian complex matrices. Let π, ν, δ be nonnegative integers with π + ν + δ = n
and denote by G (π, ν, δ) the set of all A ∈ Hn with inertia in(A) = (π, ν, δ). Assume that one of
the following is satisfied:
r n ≥ 3, k < n, φ is invertible, and rank φ(A) ≤ k whenever rank A = k.
r n ≥ 2, the range of φ is not one-dimensional, and rank φ(A) = 1 whenever rank A = 1.
r The triple (π, ν, δ) does not belong to the set {(n, 0, 0), (0, n, 0), (0, 0, n)}, φ is bijective, and
φ(G (π, ν, δ)) ⊆ G (π, ν, δ).
Then there exist an invertible n × n complex matrix R and a constant c ∈ {−1, 1} such that either
(a) φ(A) = c R AR ∗ for every A ∈ Hn or
(b) φ(A) = c R AT R ∗ for every A ∈ Hn .
Of course, if the third of the above conditions is satisfied, then the possibility c = −1 can occur
only when π = ν.
7. [BLL92, p. 76] Let n ≥ 3 and let φ : F n×n → F n×n be an invertible linear map such that φ(A)φ(B) =
φ(B)φ(A) whenever A and B commute. Then φ is an (R, c , f )-standard map for some invertible
R ∈ F n×n , some nonzero scalar c , and some linear functional f : F n×n → F .
8. [LP01, Theorem 2.3] Let φ : Cn×n → Cn×n be a linear similarity preserving map. Then either
(a) φ is an (R, c , f )-standard map or
(b) φ(A) = (tr A)B, A ∈ Cn×n .
Here, B, R ∈ Cn×n and R is invertible, c is a nonzero complex number, and f : Cn×n → C is a
functional of the form f (A) = btr A, A ∈ Cn×n , for some b ∈ C.
9. [BLL92, p. 77] Let φ be a real-linear unitary similarity preserving map on Hn . Then either
(a) φ(A) = (tr A)B, A ∈ Hn or
(b) φ(A) = cU AU ∗ + b(tr A)I , A ∈ Hn or
(c) φ(A) = cU AT U ∗ + b(tr A)I , A ∈ Hn .
Here, B is a Hermitian matrix, U is a unitary matrix, and b, c are real constants.
10. [BLL92, Theorem 4.7.6] Let n > 2 and let F be an algebraically closed field of characteristic zero.
Let p be a polynomial of degree n with at least two distinct roots. Let us write p as p(x) = x k q (x)
with k ≥ 0 and q (0) = 0. Assume that φ : F n×n → F n×n is an invertible linear map preserving the
set of all matrices annihilated by p. Then either
(a) φ(A) = c R AR −1 , A ∈ F n×n or
(b) φ(A) = c R AT R −1 , A ∈ F n×n .
Here, R is an invertible matrix and c is a constant permuting the roots of q ; that is, q (c λ) = 0
for each λ ∈ F satisfying q (λ) = 0.
11. [BLL92, p. 48] Let s l n ⊂ F n×n be the linear space of all trace zero matrices and φ : s l n → s l n
an invertible linear map preserving the set of all nilpotent matrices. Then there exist an invertible
matrix R ∈ F n×n and a nonzero scalar c such that either
(a) φ(A) = c R AR −1 , A ∈ s l n or
(b) φ(A) = c R AT R −1 , A ∈ s l n .
When considering linear preservers of nilpotent matrices one should observe first that the linear
span of all nilpotent matrices is s l n and, therefore, it is natural to confine maps under consideration
to this subspace of codimension one.
22-6
Handbook of Linear Algebra
12. [GLS00, pp. 76, 78] Let F be an algebraically closed field of characteristic 0, m, n positive integers,
and φ : F n×n → F m×m a linear transformation. If φ is nonzero and maps idempotent matrices to
idempotent matrices, then m ≥ n and there exist an invertible matrix R ∈ F m×m and nonnegative
integers k1 , k2 such that 1 ≤ k1 + k2 , (k1 + k2 )n ≤ m and
φ(A) = R(A ⊕ . . . ⊕ A ⊕ AT ⊕ . . . ⊕ AT ⊕ 0)R −1
for every A ∈ F n×n . In the above block diagonal direct sum the matrix A appears k1 times, AT
appears k2 times, and 0 is the zero matrix of the appropriate size (possibly absent). If p ∈ F [X]
is a polynomial of degree > 1 with simple zeros (each zero has multiplicity one), φ is unital and
maps every A ∈ F n×n satisfying p(A) = 0 into some m × m matrix annihilated by p, then φ is of
the above described form with (k1 + k2 )n = m.
13. [BLL92, Theorem 4.6.2] Let φ : Cn×n → Cn×n be a linear map preserving the unitary group. Then
φ is a (U, V )-standard map for some unitary matrices U, V ∈ Cn×n .
14. [KH92] Let φ : Cn×n → Cn×n be a linear map preserving normal matrices. Then either
(a) φ(A) = cU AU ∗ + f (A)I , A ∈ Cn×n or
(b) φ(A) = cU At U ∗ + f (A)I , A ∈ Cn×n or
(c) the range of φ is contained in the set of normal matrices.
Here, U is a unitary matrix, c is a nonzero scalar, and f is a linear functional on Cn×n .
15. [LP01, p. 595] Let · be a√unitarily invariant norm on Cm×n that is not a multiple of the Frobenius
norm defined by A = tr (AA∗ ). The group of linear preservers of · on Cm×n is the group
of all (U, V )-standard maps, where U ∈ Cm×m and V ∈ Cn×n are unitary matrices. Of course, if
· is a mulitple of the Frobenious norm, then the group of linear preservers of · on Cm×n is
the group of all unitary operators, i.e., those linear operators φ : Cm×n → Cm×n that preserve the
usual inner product A, B = tr (AB ∗ ) on Cm×n .
16. [BLL92, p. 63–64] Let φ : Cn×n → Cn×n be a linear map preserving the numerical radius. Then
either
(a) φ(A) = cU AU ∗ , A ∈ Cn×n or
(b) φ(A) = cU AT U ∗ , A ∈ Cn×n .
Here, U is a unitary matrix and c a complex constant with |c | = 1.
17. [BLL92, Theorem 4.3.1] Let n > 2 and let φ : F n×n → F n×n be a linear map preserving the
permanent. Then φ is an (R, S)-standard map, where R and S are each a product of a diagonal and
a permutation matrix, and the product of the two diagonal matrices has determinant one.
18. [CL98] Let φ : Tn → Tn be a linear rank one preserver. Then either
(a) The range of φ is the space of all matrices of the form
⎡
∗
⎢
⎢0
⎢
⎢.
⎢.
⎣.
0
⎤
∗
...
∗
0
..
.
...
0⎥
⎥
.. ⎥
⎥
.⎦
0
...
..
⎥
.
0
or
(b) The range of φ is the space of all matrices of the form
⎡
0
⎢
⎢0
⎢
⎢.
⎢.
⎣.
0
or
⎤
0
...
0
∗
0
..
.
...
.
0
..
.
∗⎥
⎥
.. ⎥
⎥
.⎦
0
...
0
∗
..
⎥
22-7
Linear Preserver Problems
(c) φ(A) = R AS for some invertible R, S ∈ Tn or
(d) φ(A) = R A f S for some invertible R, S ∈ Tn .
Examples:
1. Let n ≥ 2. Then the linear map φ : Tn → Tn defined by
⎛⎡
a12
...
a1n
0
..
.
a22
..
.
...
a2n ⎥⎟
⎥⎟
⎟
.. ⎥
⎥⎟
. ⎦⎠
0
0
...
⎜⎢
⎜⎢
⎜⎢
φ ⎜⎢
⎜⎢
⎝⎣
⎡
⎤⎞
a11
..
.
⎥⎟
ann
a11 + a22 + . . . + ann
a12 + a23 + . . . + an−1,n
...
a1n
0
..
.
0
..
.
...
.
0
..
.
0
0
...
0
⎢
⎢
⎢
=⎢
⎢
⎣
..
⎤
⎥
⎥
⎥
⎥
⎥
⎦
is an example of a singular preserver of rank one.
2. The most important example of a nonstandard linear preserver problem is the problem of characterizing linear maps on n × n real or complex matrices preserving the set of positive semidefinite matrices. Let R1 , . . . , Rr , S1 , . . . , Sk be n × n matrices. Then the linear map φ given by
φ(A) = R1 AR1∗ + · · · + Rr ARr∗ + S1 AT S1∗ + · · · + Sk AT Sk∗ is a linear preserver of positive
semidefinite matrices. Such a map is called decomposable. In general it cannot be reduced to a
single congruence or a single congruence composed with the transposition. Moreover, there exist
linear maps on the space of n × n matrices preserving positive semidefinite matrices that are not
decomposable. There is no general structural result for such maps.
22.4
Additive, Multiplicative, and Nonlinear Preservers
Definitions:
A map φ : F m×n → F m×n is additive if φ(A + B) = φ(A) + φ(B), A, B ∈ F m×n . An additive map
φ : F m×n → F m×n having a certain preserving property is called an additive preserver.
A map φ : F n×n → F n×n is multiplicative if φ(AB) = φ(A)φ(B), A, B ∈ F n×n . A multiplicative map
φ : F n×n → F n×n having a certain preserving property is called a multiplicative preserver.
Two matrices A, B ∈ F m×n are said to be adjacent if rank (A − B) = 1.
A map φ : F n×n → F n×n is called a local similarity if for every A ∈ F n×n there exists an invertible
R A ∈ F n×n such that φ(A) = R A AR −1
A .
Let f : F → F be an automorphism of the field F . A map φ : F n×n → F n×n defined by φ([ai j ]) =
[ f (ai j )] is called a ring automorphism of F n×n induced by f .
Facts:
1. [BS00] Let n ≥ 2 and assume that φ : F n×n → F n×n is a surjective additive map preserving rank
one matrices. Then there exist a pair of invertible matrices R, S ∈ F n×n and an automorphism f
of the field F such that φ is a composition of an (R, S)-standard map and a ring automorphism of
F n×n induced by f .
2. [GLR03] Let S L (n, C) denote the group of all n × n complex matrices A such that det A = 1.
A multiplicative map φ : S L (n, C) → Cn×n satisfies ρ(φ(A)) = ρ(A) for every A ∈ S L (n, C) if
and only if there exists S ∈ S L (n, C) such that either
22-8
Handbook of Linear Algebra
(a) φ(A) = S AS −1 , A ∈ S L (n, C) or
(b) φ(A) = S AS −1 , A ∈ S L (n, C).
Here, A denotes the matrix obtained from A by applying the complex conjugation entrywise.
3. [PS98] Let n ≥ 3 and let φ : Cn×n → Cn×n be a continuous mapping. Then φ preserves spectrum,
commutativity, and rank one matrices (no linearity, additivity, or multiplicativity is assumed) if
and only if there exists an invertible matrix R ∈ Cn×n such that φ is an (R, R −1 )-standard map.
4. [BR00] Let φ : Cn×n → Cn×n be a spectrum preserving C 1 -diffeomorphism (again, we do not
assume that φ is additive or multiplicative). Then φ is a local similarity.
5. [HHW04] Let n ≥ 2. Then φ : Hn → Hn is a bijective map such that φ(A) and φ(B) are adjacent
for every adjacent pair A, B ∈ Hn if and only if there exist a nonzero real number c , an invertible
R ∈ Cn×n , and S ∈ Hn such that either
(a) φ(A) = c R AR ∗ + S, A ∈ Hn or
(b) φ(A) = c R AR ∗ + S, A ∈ Hn .
6. [Mol01] Let n ≥ 2 be an integer and φ : Hn → Hn a bijective map such that φ(A) ≤ φ(B) if and
only if A ≤ B, A, B ∈ Hn (here, A ≤ B if and only if B − A is a positive semidefinite matrix).
Then there exist an invertible R ∈ Cn×n and S ∈ Hn such that either
(a) φ(A) = R AR ∗ + S, A ∈ Hn or
(b) φ(A) = R AR ∗ + S, A ∈ Hn .
This result has an infinite-dimensional analog important in quantum mechanics. In the language
of quantum mechanics, the relation A ≤ B means that the expected value of the bounded observable
A in any state is less than or equal to the expected value of B in the same state.
Examples:
1. We define a mapping φ : Cn×n → Cn×n in the following way. For a diagonal matrix A with distinct
diagonal entries, we define φ(A) to be the diagonal matrix obtained from A by interchanging the
first two diagonal elements. Otherwise, let φ(A) be equal to A. Clearly, φ is a bijective mapping
preserving spectrum, rank, and commutativity in both directions. This shows that the continuity
assumption is indispensable in Fact 3 above.
2. Let φ : Cn×n → Cn×n be a map defined by φ(0) = E 12 , φ(E 12 ) = 0, and φ(A) = A for all
A ∈ Cn×n \ {0, E 12 }. Then φ is a bijective spectrum preserving map that is not a local similarity.
More generally, we can decompose Cn×n into the disjoint union of the classes of matrices having
the same spectrum and then any bijection leaving each of this classes invariant preserves spectrum.
Thus, the assumption on differentiability is essential in Fact 4 above.
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