In mathematics, an EP matrix (or range-Hermitian matrix
or RPN matrix
) is a square matrix ''A'' whose range is equal to the range of its
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
''A''*. Another equivalent characterization of EP matrices is that the range of ''A'' is orthogonal to the nullspace of ''A''. Thus, EP matrices are also known as RPN (Range Perpendicular to Nullspace) matrices.
EP matrices were introduced in 1950 by
Hans Schwerdtfeger
Hans Wilhelm Eduard Schwerdtfeger (9 December 1902 – 26 June 1990) was a German-Canadian-Australian mathematician who worked in Galois theory, matrix theory, theory of groups and their geometries, and complex analysis.
"In 1962 he publish ...
,
and since then, many equivalent characterizations of EP matrices have been investigated through the literature.
The meaning of the EP abbreviation stands originally for ''E''qual ''P''rincipal, but it is widely believed that it stands for ''Equal
Projectors
A projector or image projector is an optical device that projects an image (or moving images) onto a surface, commonly a projection screen. Most projectors create an image by shining a light through a small transparent lens, but some newer type ...
'' instead, since an equivalent characterization of EP matrices is based in terms of equality of the projectors ''AA
+'' and ''A
+A''.
The range of any matrix ''A'' is perpendicular to the null-space of ''A''*, but is not necessarily perpendicular to the null-space of ''A''. When ''A'' is an EP matrix, the range of ''A'' is precisely perpendicular to the null-space of ''A''.
Properties
* An equivalent characterization of an EP matrix ''A'' is that ''A'' commutes with its
Moore-Penrose inverse, that is, the projectors ''AA
+'' and ''A
+A'' are equal. This is similar to the characterization of normal matrices where ''A'' commutes with its conjugate transpose.
As a corollary,
nonsingular matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
are always EP matrices.
* The sum of EP matrices ''A''
i is an EP matrix if the null-space of the sum is contained in the null-space of each matrix ''A''
i.
* To be an EP matrix is a necessary condition for normality: ''A'' is normal if and only if ''A'' is EP matrix and ''AA''*''A''
2 = ''A''
2''A''*''A''.
* When ''A'' is an EP matrix, the Moore-Penrose inverse of ''A'' is equal to the
group inverse of ''A''.
* ''A'' is an EP matrix if and only if the Moore-Penrose inverse of ''A'' is an EP matrix.
Decomposition
The
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful b ...
states that a matrix is normal if and only if it is
unitarily similar to a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
.
Weakening the normality condition to EPness, a similar statement is still valid. Precisely, a matrix ''A'' of rank ''r'' is an EP matrix if and only if it is unitarily similar to a core-nilpotent matrix,
that is,
:
where ''U'' is an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ...
and ''C'' is an ''r'' x ''r'' nonsingular matrix. Note that if ''A'' is full rank, then ''A'' = ''UCU''
*.
References
{{Reflist
Matrices