Let A be an n-by-n matrix. The numerical range of A is defined as . The Moore–Penrose inverse of A is the unique matrix satisfying , and . This paper investigates the numerical range of the Moore–Penrose inverse of a matrix A, and examines the relation between the numerical ranges and .
Let , the complex matrices, the Moore–Penrose inverse is the unique matrix that satisfies the following properties [1,2]:
Consider the system of linear equations:
Moore and Penrose showed that is a vector x such that is minimized among all vectors x for which is minimal. The theory and applications of the Moore–Penrose inverse can be found, for examples, in [3,4,5].
Let be the set of complex matrices. The numerical range of is defined as
The numerical radius of A is defined by the identity . The well-known Toeplitz–Hausdorff theorem asserts that is a convex set containing the spectrum of A. There are several fundamental facts about the numerical ranges of square matrices:
, U unitary;
, where is the direct sum of and ;
if and only if A is Hermitian;
If A is normal then is the convex of .
(For references on the numerical range and its generalizations, see, for instance, ref. ).
The numerical range of of a nonsingular matrix is developed in [7,8] for which the spectrum of any matrix is characterized as the intersection of a family of the numerical ranges of the inverses of nonsingular matrices. In this paper, we investigate the numerical ranges of the Moore–Penrose inverses, and examine the relationship of the numerical ranges between and . In particular, we prove in Section 2 that if and only if , and
Recall that the singular value decomposition of a matrix with rank k is written as , where and are unitary, and has for all , and , . The entries are called the singular values of A (cf. ). The following facts list a number of useful properties concerning the Moore–Penrose inverse.
Assume is a singular value decomposition of A, then .
If is nonsingular, .
If , , then .
For any nonzero vector , .
If , for any unitary matrices and , .
Throughout this paper, we define if .
2. Numerical Range
We begin with two examples to observe some properties of the geometry between the numerical ranges and .
Consider a rank one matrix
By the singular value decomposition of A, we find that
where . Clearly, both and are elliptic disks.
On the other hand, the following example indicates that and may differ in geometry types.
Let . Consider the matrix
By (F3), and taking in Example 1, we have
Then is a polygon, but is an elliptic disk.
The following result can be easily derived from facts (F5) and (F3).
Let . Then A is normal (resp. hermitian) if and only if is normal (resp. hermitian).
Theorem 1 asserts that both and have the same geometry type, namely convex polygons or line segments, depending on A is normal or hermitian. We show in Theorem 1 that certain non-normal matrices also admit this property.
The following result shows that the spectra of A and as well as their numerical ranges simultaneously contain the origin.
Let . Then
if and only if .
if and only if .
If A is normal and then if and only if .
By the properties and we have if and only if . This proves .
Suppose A is singular. Then, by , if and only if . Suppose A is nonsingular. Then , and
for some if and only if
if and only if for some , which is equivalent to This proves .
If A is normal with spectrum decomposition , then . Suppose the diagonal matrix , . It is easy to see that , and thus follows. □
Choose in Example 1. It shows that of Theorem 2 may fail for non-normal matrices.
As a consequence of Theorem 2, we obtain the following reciprocal convexity.
Let be nonzero complex numbers. If
for some nonnegative with , then there exist nonnegative with such that
Consider the diagonal matrix . If , then , the convex polygon with vertices . By Theorem 2 , we have that
which is convex polygon with vertices Therefore, there exist nonnegative with such that
Let . If is symmetric with respect to x-axis then
for every singular value s of A.
Let be a singular value decomposition of A, where If is a singular value of A, then A is singular. Hence , and thus .
If is a nonzero singular value of A, we may assume , then 1 is a singular value of Choose a unit vector x such that with only nonzero first coordinate. Then Since is symmetric with respect to x-axis, Hence
On the other hand, Then
which is equivalent to
The result of Theorem 4 may fail if the symmetric property of the numerical range of A is omitted. For example, consider the matrix
Then the singular values of A are , and
In this case, for every singular value , we have that
It is mentioned in [7,8,10], for any nonsingular matrix ,
We present the spectrum inclusion in Equation (1) for Moore–Penrose inverses.
Let . If then
It is well known that . Suppose . If then , and by of Theorem 2, . The inclusion in Equation (2) holds. Assume . Choose a unit eigenvector x with . Then
A matrix satisfying the condition in Theorem 5 is called an EP matrix. Baksalary  proposed that the class of EP matrices is characterized as those matrices A for which the column space of coincides with the column space of . Bapat et al. , confirmed the characterization. The EP assumption in Theorem 5 is essential. For instance, taking in Example 1, then the eigenvalue 1 of A is not in since . Note that and are even unitarily equivalent.
It is shown in , under rank additivity , the Moore–Prnrose inverse can be represented in terms of and . Applying the result, there obtains
for any orthonormal vectors . We extend Equation (7) to a general result.
Let and be two orthonormal subsets of . If then , and .
Extend and to orthonormal bases and of , respectively. Let and be the corresponding unitary matrices. Then
Hence, by (F5),
It follows that , and thus . □
3. Bounds on Numerical Radii
Recall that for any nonsingular matrix A, the number is called the condition number of A with respect to the given matrix norm. The matrix A is ill conditioned if its condition number is large.
For any matrix A, nonsingular or not, we also call the number the condition number of the matrix A.
Let . Then, for the spectral norm ,
If , there exists x such that . Then , . Since is idempotent and hermitian, it follows that . Thus, . By the numerical radius inequality (cf.  p. 44), we obtain that
Let be a weighted shift matrix
It is well known that is a circular disk centered at the origin. The radius of the circle has attracted the attention of many authors, see for example, refs. [14,15,16,17]. In particular, if , (cf. [15,17]). For weighted shift matrices, upper bounds of the numerical radii are found in [14,16]. The Moore–Penrose inverse provides an upper bound and a lower bound for the numerical radii of certain weighted shift matrices.
Let be a weighted shift matrix defined by Equation (8). Then
and are circular disks centered at the origin, and
where the minimum is taken over those k with .
If for all then , and
Assume a singular value decomposition of A is
and Direct computations on obtain the representation in Equation (9) of . It is easy to see that in Equation (9) is permutationally equivalent to the weighted shift matrix
The circularity of and follows a well known result that the numerical range of any weighted shift matrix is a circular disk centered at the origin (cf. ), and the numerical range of the transpose of a matrix equals the numerical range of the matrix itself. Moreover, by Theorem 3 in , the numerical radius
Together with Theorem 7, the assertion follows.
If for all , then is permutationally equivalent to the matrix in Equation (10) which is exactly equal to A. Thus Suppose . Then , and the numerical radius inequality follows from □
The lower bound in is sharp as can be easily seen by taking and
This work was partially supported by Ministry of Science and Technology, Taiwan, under NSC 99-2115-M-031-004-MY2.
The author thanks the referees for their helpful comments and suggestions on earlier versions.
Conflicts of Interest
The author declares no conflict of interest.
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