Citation: | LI Zhen. Schur Forms and Normal-Nilpotent Decompositions[J]. Applied Mathematics and Mechanics, 2024, 45(9): 1200-1211. doi: 10.21656/1000-0887.450129 |
In the thesis of Li[1], a normal-nilpotent decomposition (NND) of real square matrices is proposed and applied to the velocity gradient tensor of a fluid element for analyzing vortex criteria (also see ref. [2]). The decomposition can be stated as that every 3-by-3 real matrix is a sum of a normal matrix and a nilpotent matrix. As a special application, the velocity gradient tensor
$$ {\boldsymbol{A}} = {\boldsymbol{N}} + {\boldsymbol{S}}, $$ | (1) |
where
The NND proposed in ref. [1-2] is derived from a canonical form of real matrices discovered earlier by Murnaghan, Wintner[3], which is a counterpart over the field of real numbers to the famous Schur form for complex matrices[4]. In that paper, Isaai Schur generalized the previous results restricted to orthogonal[5] and Hermitian[6] matrices. The canonical form of Murnaghan, Wintner[3] is often referred to as the real Schur form. It is known that both the complex and real Schur forms of a given matrix are not unique. Actually, the NND is based on a special real Schur form among the possible variants. However, ref. [1-2] didn’t present a clarification of which variant of real Schur forms has been chosen and how to maintain uniqueness and consistency throughout the flow fields.
Recently, real and complex Schur forms as well as NND have received increasing attention. For example, Keylock[7-8] employed the complex Schur form of velocity gradient tensor in their statistical model of fine structures in turbulence. Das, Girimaji[9] analyzed fine structures of turbulence in the framework of NND. Hoffman[10] used NND to analyze energy stability in incompressible turbulence. Zhu[11-12] derived a real Schur flow (RSF) as the compressible Taylor-Proudman limit in magnetohydrodynamics, for which the velocity gradient tensor takes a globally uniform real Schur form. Kronborg et al. [13] applied NND to analyse the shear in blood flows. Arun, Colonius[14] analyze the collision of vortex rings with NND. Just to mention a few.
Apart from the direct application of Schur forms and NND as mentioned above, there are also studies of fluid motions that are related to or similar in some sense to NND. For example, Kolář[15-16] proposed a triple decomposition of motion (TDM) of the velocity gradient tensor, which is very similar to the quadruple NND proposed in ref. [1-2]. Unfortunately, several works, such as ref. [9-10, 13-14], misidentified the tools they used as TDM, when in fact they were NND.
Kronborg, Hoffman[17] made an effort to clarify the relationship between TDM and the Schur forms. But they overlooked the gap between the two; especially, they misidentified NND as TDM. Also, they didn’t reveal the intrinsic gap between the NNDs derived from real and complex Schur forms properly, and treated it merely as an issue of algorithmic convenience. Besides, they failed to characterize the non-uniqueness of these forms completely and to present a satisfactory standardization procedure, although they claimed so.
Considering the different types and non-uniqueness of Schur forms as well as the various possible definitions of NNDs, a clarification of their relationships is in demand, which is the aim of the current article. We hope that this article will help eliminate the confusion that has existed in previous research and clear the way for further investigations and applications of Schur forms and NNDs in fluid mechanics and other areas. The readers may also have an interest in the geometric or kinematical interpretations of Schur forms, which is out of the scope of this article. We refer to Zou, Xu, Tang[18], which provides a nice discussion on streamline patterns of real Schur forms.
The structure of the article is as follows. Since NND is based on a real Schur form, we first describe the Schur forms. In section 1, the general complex Schur form for a 3-by-3 real matrix is derived with a discussion of its non-uniqueness. In section 2, the general real Schur form for a 3-by-3 real matrix is derived with a discussion of its non-uniqueness. A condition of uniqueness and the special real Schur form determined by this condition are proposed. In section 3, a brief discussion of normal and nilpotent matrices are presented firstly. Based on the general complex Schur form, a construction of complex NND is presented. Based on the special real Schur form proposed in section 2, several types of real NNDs are proposed. Besides, we also proposed two normal-nonnormal decompositions. Then, the distinction between the NNDs and the TDM proposed by Kolář[15-16] is demonstrated. In section 4, we demonstrate that it is generally impossible to convert a complex NND into a real NND through unitary transformations, which implies that the two types of NNDs are nonequivalent.
For later use, let’s first restate a classical result about matrices. Let
$$ {\boldsymbol{A}} = {\boldsymbol{U}}^\dagger \tilde{{\boldsymbol{A}}} {\boldsymbol{U}} = \sum\limits_{i,j:i\le j} \tilde{A}_{ij} {\boldsymbol{u}}_i^\dagger {\boldsymbol{u}}_j . $$ | (2) |
The eq.(2) is sometimes stated as a theorem and referred to as the representation theorem of matrices.
Theorem 1 Every matrix
Indeed, it is an extension of the representation theorem of self-adjoint matrices, which says that every self-adjoint matrix has a diagonal representation in certain orthonormal basis. An immediate consequence of the theorem is
Corollary 1 In the representation formula eq.(2),
Since
Let
Fix
$${\boldsymbol{U}}:=\left(\begin{matrix}{\boldsymbol{u}} \\ {\boldsymbol{v}} \\ {\boldsymbol{e}}_{{\rm{r}}}\end{matrix}\right)$$ | (3) |
is a unitary matrix. There is
$$ \tilde{{\boldsymbol{A}}} := {\boldsymbol{U}} {\boldsymbol{A}} {\boldsymbol{U}}^\dagger = \left(\begin{matrix} {\tilde{{\boldsymbol{A}}}_2 }& { {\boldsymbol{c}}^\dagger }\\ { {\bf{0}}} & {\lambda_{ {\rm{r}}} } \end{matrix}\right) $$ | (4) |
with submatrices (i.e., blocks)
$$\tilde{{\boldsymbol{A}}}_2 := \left(\begin{matrix}{{\boldsymbol{u}} {\boldsymbol{A}} {\boldsymbol{u}}^\dagger} & {{\boldsymbol{u}} {\boldsymbol{A}} {\boldsymbol{v}}^\dagger }\\ { {\boldsymbol{v}} {\boldsymbol{A}} {\boldsymbol{u}}^\dagger} & {{\boldsymbol{v}} {\boldsymbol{A}} {\boldsymbol{v}}^\dagger } \end{matrix}\right),\; {\boldsymbol{c}}^\dagger := \left(\begin{matrix} { {\boldsymbol{u}} {\boldsymbol{A}} {\boldsymbol{e}}_{ {\rm{r}}}^\dagger }\\ { {\boldsymbol{v}} {\boldsymbol{A}} {\boldsymbol{e}}_{ {\rm{r}}}^\dagger } \end{matrix}\right), $$ | (5) |
where
$$ {\boldsymbol{V}}:= \left(\begin{matrix} { {\boldsymbol{V}}_2 }& { {\bf{0}}} \\ { {\bf{0}} }&{ 1 } \end{matrix}\right) $$ | (6) |
will do the equivalent job as
Besides of
Let
$$ {\boldsymbol{V}}_2 = \left(\begin{matrix} {\boldsymbol{a}} \\ {\boldsymbol{b}} \end{matrix}\right) \; ,$$ | (7) |
with
$$\tilde{\tilde{{\boldsymbol{A}}}}_2 := {\boldsymbol{V}}_2 \tilde{{\boldsymbol{A}}}_2 {\boldsymbol{V}}_2^\dagger = \left(\begin{matrix} {{\boldsymbol{a}} \tilde{{\boldsymbol{A}}}_2 {\boldsymbol{a}}^\dagger }& {{\boldsymbol{a}} \tilde{{\boldsymbol{A}}}_2 {\boldsymbol{b}}^\dagger} \\{{\boldsymbol{b}} \tilde{{\boldsymbol{A}}}_2 {\boldsymbol{a}}^\dagger }& { {\boldsymbol{b}} \tilde{{\boldsymbol{A}}}_2 {\boldsymbol{b}}^\dagger } \end{matrix}\right). $$ | (8) |
If
Being an upper triangular matrix,
$$\tilde{\tilde{{\boldsymbol{A}}}} := {\boldsymbol{V}} \tilde{{\boldsymbol{A}}} {\boldsymbol{V}}^\dagger = \left(\begin{matrix} { \tilde{\tilde{{\boldsymbol{A}}}}_2 }& { {\boldsymbol{V}}_2 {\boldsymbol{c}}^\dagger }\\ { {\bf{0}} }& {\lambda_{ {\rm{r}}} } \end{matrix}\right) = \left(\begin{matrix} { \lambda_1 }& {{\boldsymbol{a}} \tilde{{\boldsymbol{A}}}_2 {\boldsymbol{b}}^\dagger} & { {\boldsymbol{a}} {\boldsymbol{c}}^\dagger }\\ { 0} & {\lambda_2} & { {\boldsymbol{b}} {\boldsymbol{c}}^\dagger} \\ { 0 }&{ 0} & {\lambda_{\rm{r}} } \end{matrix}\right), $$ | (9) |
which is a complex Schur form of
$$ {\boldsymbol{P}} := {\rm{diag}}\,({{\rm{e}}^{ {\rm{i}} \phi_1},{\rm{e}}^{ {\rm{i}} \phi_2},{\rm{e}}^{ {\rm{i}} \phi_3}}).$$ | (10) |
Then
$$ \tilde{\tilde{{\boldsymbol{A}}}} = \left(\begin{matrix}{ \lambda_1 } &{ a_{12} {\rm{e}}^{ {\rm{i}} (\phi_1-\phi_2)} } &{ a_{13} {\rm{e}}^{ {\rm{i}} (\phi_1-\phi_3)} }\\ { 0 } &{ \lambda_2 } &{ a_{23} {\rm{e}}^{ {\rm{i}} (\phi_2-\phi_3)} }\\ { 0 } &{ 0 } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right). $$ | (11) |
Notice that only the differences between
$$ \tilde{\tilde{{\boldsymbol{A}}}} = \left(\begin{matrix} { \lambda_1 } &{ a_{12} {\rm{e}}^{ {\rm{i}} (\phi_{13}-\phi_{23})} } &{ a_{13} {\rm{e}}^{ {\rm{i}} \phi_{13}} }\\ { 0 } &{ \lambda_2 } &{ a_{23} {\rm{e}}^{ {\rm{i}} \phi_{23}} }\\ { 0 } &{ 0 } &{ \lambda_3} \end{matrix}\right). $$ | (12) |
Since each of
If
$$ \tilde{\tilde{{\boldsymbol{A}}}} = \left(\begin{matrix} { \lambda_1 } &{ a_{12} (-1)^{m-n} } &{ a_{13} (-1)^m }\\ { 0 } &{ \lambda_2 } &{ a_{23} (-1)^n }\\ { 0 } &{ 0 } &{ \lambda_3 } \end{matrix}\right). $$ | (13) |
This is the real Schur form of
As mentioned at the end of section 1, When
$$ {\boldsymbol{Q}} = \left(\begin{matrix} { \cos\,\varrho } &{ \sin\,\varrho } &{ 0 }\\ { -\sin\,\varrho } &{ \cos\,\varrho } &{ 0 }\\ { 0 } &{ 0 } &{ 1 } \end{matrix}\right), $$ | (14) |
then
$$ \hat{{\boldsymbol{A}}} := {\boldsymbol{Q}} \tilde{{\boldsymbol{A}}} {\boldsymbol{Q}}^ {\rm{T}}= $$ | (15) |
$$\qquad \left(\begin{matrix} {\begin{array}{*{20}{c}}{\tilde{A}_{11} \cos ^2 \varrho+\tilde{A}_{22} \sin\, \varrho+}\\{(\tilde{A}_{12}+\tilde{A}_{21}) \cos \,\varrho \sin \,\varrho}\end{array}} & {\begin{array}{*{20}{c}}{\tilde{A}_{12} \cos ^2 \varrho-\tilde{A}_{21} \sin ^2 \varrho+}\\{(\tilde{A}_{22}-\tilde{A}_{11}) \cos \,\varrho \sin \,\varrho}\end{array}} & {c_1 \cos \,\varrho+c_2 \sin \,\varrho} \\ {\begin{array}{*{20}{c}}{ \tilde{A}_{21} \cos ^2 \varrho-\tilde{A}_{12} \sin ^2 \varrho+}\\{ (\tilde{A}_{22}-\tilde{A}_{11}) \cos \,\varrho \sin \,\varrho} \end{array}}& {\begin{array}{*{20}{c}}{\tilde{A}_{22} \cos ^2 \varrho+\tilde{A}_{11} \sin ^2 \varrho-}\\{(\tilde{A}_{12}+\tilde{A}_{21}) \cos \,\varrho \sin \,\varrho} \end{array}}& {{c_2 \cos \,\varrho-c_1 \sin \,\varrho }}\\ {0} & {0} & {\lambda_{{\rm{r}}}}\end{matrix}\right).$$ | (16) |
We require that
$$ ( \tilde{A}_{11}- \tilde{A}_{22})({\cos^2\varrho - \sin^2\varrho}) + 2( \tilde{A}_{12}+ \tilde{A}_{21})\cos\,\varrho\sin\,\varrho = 0, $$ | (17) |
which implies that
$$ \tan\,(2\varrho) = \frac{ \tilde{A}_{22}- \tilde{A}_{11}}{ \tilde{A}_{12}+ \tilde{A}_{21}}. $$ | (18) |
Since the period of
$$ \cos^2\varrho = \frac{1}{2}\left({1\pm\frac{ \tilde{A}_{12}+ \tilde{A}_{21}}{| \tilde{{\boldsymbol{A}}}_2|}}\right),\;\; \sin\,\varrho\cos\,\varrho = \pm\frac{ \tilde{A}_{22}- \tilde{A}_{11}}{2| \tilde{{\boldsymbol{A}}}_2|}, $$ | (19) |
where
$$ | \tilde{{\boldsymbol{A}}}_2|:=\sqrt{\|{ \tilde{{\boldsymbol{A}}}_2}\|_F^2-2\det \tilde{{\boldsymbol{A}}}_2}=\sqrt{({ \tilde{A}_{12}+ \tilde{A}_{21}})^2 + ({ \tilde{A}_{22}- \tilde{A}_{11}})^2} $$ | (20) |
is invariant under unitary transformations, hence the components of
$$ \hat{A}_{11} = \hat{A}_{22} = \frac{ \tilde{A}_{11}+ \tilde{A}_{22}}{2},\;\; \hat{A}_{12} = \frac{ \tilde{A}_{12}- \tilde{A}_{21}\pm| \tilde{{\boldsymbol{A}}}_2|}{2}, $$ | (21) |
$$ \hat{A}_{21} = \frac{ \tilde{A}_{21}- \tilde{A}_{12}\pm| \tilde{{\boldsymbol{A}}}_2|}{2}. $$ | (22) |
Therefore,
$$ \hat{{\boldsymbol{A}}} = \left(\begin{matrix} { \dfrac{ \tilde{A}_{11}+ \tilde{A}_{22}}{2} } &{ \dfrac{ \tilde{A}_{12}- \tilde{A}_{21}\pm| \tilde{{\boldsymbol{A}}}_2|}{2} } &{ \hat{A}_{13} }\\ { \dfrac{ \tilde{A}_{21}- \tilde{A}_{12}\pm| \tilde{{\boldsymbol{A}}}_2|}{2} } &{ \dfrac{ \tilde{A}_{11}+ \tilde{A}_{22}}{2} } &{ \hat{A}_{23}} \\ { 0 } &{ 0 } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right). $$ | (23) |
Denote
$$ \chi := \hat{A}_{11} = \hat{A}_{22},\;\; \omega_3:= \hat{A}_{12}- \hat{A}_{21},\;\; \gamma := \hat{A}_{12} + \hat{A}_{21} = \pm| \hat{{\boldsymbol{A}}}_2|. $$ | (24) |
The 2-by-2 diagonal block of
$$ \hat{{\boldsymbol{A}}}_2 := \left(\begin{matrix} { \chi} &{ \dfrac{\gamma + \omega_3}{2}} \\ { \dfrac{\gamma - \omega_3}{2}} & {\chi } \end{matrix}\right) . $$ | (25) |
We have mentioned that
$$ \hat{A}_{13} := -\beta, \;\; \hat{A}_{23} := \alpha. $$ | (26) |
The two choices of
$$ \hat{{\boldsymbol{A}}} = \left(\begin{matrix} { \chi } &{ \dfrac{\gamma + \omega_3}{2} } &{ -\beta }\\ { \dfrac{\gamma - \omega_3}{2} } &{ \chi } &{ \alpha }\\ { 0 } &{ 0 } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right) $$ | (27) |
can be uniquely determined by requiring
$$ \omega_3>0,\; \gamma>0,\; \alpha>0. $$ | (28) |
Of course, this uniqueness condition is not the only choice. For example, one can require
When
$$ \lambda_{ {\rm{c}} {\rm{r}}} = \chi,\; \lambda_{ {\rm{c}} {\rm{i}}} =\frac{1}{2} \sqrt{\omega_3^2-\gamma^2}. $$ | (29) |
We should emphasize that when
$$ \lambda_{1,2} = \chi \pm \frac{1}{2} \sqrt{\gamma^2-\omega_3^2}. $$ | (30) |
Remark 1 In eq.(27), notice that
$$ 4 \hat{A}_{12} \hat{A}_{21} = \gamma^2 - \omega_3^2 $$ | (31) |
is the discriminant of the characteristic polynomial of
Remark 2 Since
A matrix
There is another set of necessary and sufficient conditions for normal matrices, which may serve as guidance when constructing NNDs. Let
$$ {\boldsymbol{S}} = \frac{ {\boldsymbol{A}}+ {\boldsymbol{A}}^\dagger}{2},\; {\boldsymbol{W}} = \frac{ {\boldsymbol{A}}- {\boldsymbol{A}}^\dagger}{2}. $$ | (32) |
Then
$$ {\boldsymbol{S}} = \left(\begin{matrix} {s_1 } &{ 0 } &{ 0 }\\{0 } &{ s_2 } &{ 0 }\\{0 } &{ 0 } &{ s_3 } \end{matrix}\right),\; {\boldsymbol{W}} = \left(\begin{matrix} { 0 } &{ w_3 } &{ -w_2 }\\{-w_3 } &{ 0 } &{ w_1 }\\ { w_2 } &{ -w_1 } &{ 0 } \end{matrix}\right). $$ | (33) |
Then
$$ {\boldsymbol{S}} {\boldsymbol{W}}- {\boldsymbol{W}} {\boldsymbol{S}} = \left(\begin{matrix} {0} &{ w_3(s_1-s_2)} & w_2(s_3-s_1) \\{w_3(s_1-s_2)} & {0} & {w_1(s_2-s_3) }\\{w_2(s_3-s_1) }&{ w_1(s_2-s_3)} &{ 0} \end{matrix}\right), $$ | (34) |
which leads to the following
Theorem 2 The matrix
1) All
2)
3)
Notice that the validity of these conditions does not depend on the diagonal form of
Let
$$ \tilde{\tilde{{\boldsymbol{\varLambda}}}} = \left(\begin{matrix} {\lambda_1 } &{ 0 } &{ 0 }\\ { 0 } &{ \lambda_2 } &{ 0 }\\{0 } &{ 0 } &{ \lambda_3 } \end{matrix}\right),\; \tilde{\tilde{{\boldsymbol{\varDelta}}}} = \left(\begin{matrix}{0 } &{ a_{12} {\rm{e}}^{ {\rm{i}} (\phi_{13}-\phi_{23})} } &{ a_{13} {\rm{e}}^{ {\rm{i}} \phi_{13}} }\\{0 } &{ 0 } &{ a_{23} {\rm{e}}^{ {\rm{i}} \phi_{23}}} \\{0 } &{ 0 } &{ 0 }\end{matrix}\right). $$ | (35) |
Then we have
The NND used by Keylock[7-8] is of this type. It has several advantages, such as
1) The Schur form has a simple upper triangular form with eigenvalues on its diagonal.
2) There is only one natural way of decomposing the Schur form into the sum of a normal part and a nilpotent part.
3) The normal part and the nilpotent part have no overlap, resulting in a clean decomposition.
4) The cases of real and complex conjugate eigenvalues are treated in a unified way.
As a mathematical model of physical quantities such as the velocity gradient tensor, the complex NND is adequate. However, the physical interpretation of the complex matrices is not clear yet. In contrast, real NNDs can have a much clearer physical meaning, as demonstrated in the literature.
In Li[1], the real NND of
Let’s consider the case of complex eigenvalues first. Since the special real Schur form eq.(27) has two identical diagonal entries, the normal part of
1) Quasiorthogonal-nilpotent;
2) Symmetric-nilpotent;
3) Symmetric-antisymmetric.
Each type of decomposition above leads to a type of decomposition of
Denote
$$ \phi := \frac{\omega_3 + \gamma}{2},\; \; \psi := \frac{\omega_3 - \gamma}{2}. $$ | (36) |
The decomposition
$$ \hat{{\boldsymbol{A}}} = \hat{{\boldsymbol{N}}} + \hat{{\boldsymbol{\varGamma}}} $$ | (37) |
can be realized in the following ways.
The normal component
(a) Minimize the normal part:
$$ \hat{{\boldsymbol{N}}} := \left(\begin{matrix}{ \chi } &{ \psi } &{ 0} \\{-\psi } &{ \chi } &{ 0 }\\{0 } &{ 0 } &{ \lambda_{ {\rm{r}}}}\end{matrix}\right),\; \; \hat{{\boldsymbol{\varGamma}}} := \left(\begin{matrix} {0 } &{ \gamma } &{ -\beta }\\{0 } &{ 0 } &{ \alpha }\\{0 } &{ 0 } &{ 0}\end{matrix}\right). $$ | (38) |
In this case,
(b) Maximize the normal part:
$$ \hat{{\boldsymbol{N}}} := \left(\begin{matrix} { \chi } &{ \phi } &{ 0 }\\ { -\phi } &{ \chi } &{ 0 }\\ { 0 } &{ 0 } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right),\; \; \hat{{\boldsymbol{\varGamma}}} := \left(\begin{matrix} { 0 } &{ 0 } &{ -\beta} \\ { \gamma } &{ 0 } &{ \alpha }\\ { 0 } &{ 0 } &{ 0 } \end{matrix}\right). $$ | (39) |
In this case,
The normal component
(a) Minimize the normal part:
$$ \hat{{\boldsymbol{N}}} := \left(\begin{matrix} { \chi } &{ -\psi } &{ 0} \\ { -\psi } &{ \chi } &{ 0 }\\ { 0 } &{ 0 } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right),\; \hat{{\boldsymbol{\varGamma}}} := \left(\begin{matrix}{0 } &{ \omega_3 } &{ -\beta }\\ { 0 } &{ 0 } &{ \alpha }\\ { 0 } &{ 0 } &{ 0} \end{matrix}\right). $$ | (40) |
(b) Maximize the normal part:
$$ \hat{{\boldsymbol{N}}} := \left(\begin{matrix}{\chi } &{ \phi } &{ 0} \\{\phi } &{ \chi } &{ 0} \\{0 } &{ 0 } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right),\; \hat{{\boldsymbol{\varGamma}}} := \left(\begin{matrix}{0 } &{ 0 } &{ -\beta }\\{-\omega_3 } &{ 0 } &{ \alpha }\\{0 } &{ 0 } &{ 0 } \end{matrix}\right). $$ | (41) |
The
(a) Symmetric-nonnormal:
$$ \hat{{\boldsymbol{N}}} := \left(\begin{matrix} {\chi } &{ \dfrac{\gamma}{2} } &{ 0} \\{\dfrac{\gamma}{2} } &{ \chi } &{ 0 }\\{0 } &{ 0 } &{ \lambda_{ {\rm{r}}}}\end{matrix}\right),\; \hat{{\boldsymbol{\varGamma}}} := \left(\begin{matrix}{0 } &{ \dfrac{\omega_3}{2} } &{ -\beta }\\{-\dfrac{\omega_3}{2} } &{ 0 } &{ \alpha }\\{0 } &{ 0 } &{ 0}\end{matrix}\right). $$ | (42) |
(b) Quasiorthogonal-nonnormal:
$$ \hat{{\boldsymbol{N}}} := \left(\begin{matrix} {\chi } &{ \dfrac{\omega_3}{2} } &{ 0} \\{-\dfrac{\omega_3}{2} } &{ \chi } &{ 0 }\\{0 } &{ 0 } &{ \lambda_{ {\rm{r}}} }\end{matrix}\right),\; \hat{{\boldsymbol{\varGamma}}} := \left(\begin{matrix} {0 } &{ \dfrac{\gamma}{2} } &{ -\beta} \\{\dfrac{\gamma}{2} } &{ 0 } &{ \alpha }\\{0 } &{ 0 } &{ 0}\end{matrix}\right). $$ | (43) |
Remark 3 When
$$ \chi = \frac{\lambda_1 + \lambda_2}{2} $$ | (44) |
and
Remark 4 Kolář[15-16] defined a triple decomposition of motion (TDM) earlier than NND of Li[1], but the two decompositions have very similar appearances, which has caused some confusion. Here we point out several distinctions between the two:
1) The purely asymmetric tensor in ref. [15-16], defined by the vanishing of the product of each pair of off-diagonal entries, is generally not nilpotent. For example, the matrix
$$ \left(\begin{matrix}{ 0 } &{ \gamma } &{ 0} \\{0 } &{ 0 } &{ \alpha }\\{\beta } &{ 0 } &{ 0}\end{matrix}\right) $$ | (45) |
is purely asymmetric according to ref. [15-16], but its determinant is nonzero, indicating nonzero eigenvalues. Hence it is not nilpotent. Of course, the purely asymmetric tensor can be nilpotent in several special cases, especially in the 2D case. But the concept behind it is different from the nilpotent tensor. Actually, a condition of nilpotency can be phrased as a given
2) TDM maximizes the norm of the purely asymmetric tensor through changing of basis, while the NNDs do not maximize the nilpotent tensor, neither do they choose basis according to optimization principles.
3) Regardless of the difference in their choice of basis, TDM and NND produce different results. For example, suppose the basic reference frame (BRF) of ref. [15-16] is the same as the basis of the special real Schur form eq.(27). When
In this section, we show that it is generally impossible to transform a complex NND into a real NND. Consider the general complex Schur form eq.(12). Its complex NND is just the splitting of the diagonal and off-diagonal parts, as given by eq.(35). Let’s find a unitary transformation that converts it to a real NND. Since
Denote
$$ {\boldsymbol{\varLambda}}_2 := {\rm{diag}}\,\left({\lambda_1,\lambda_2}\right), $$ | (46) |
where
$$ {\boldsymbol{R}}_2 := \frac{1}{\sqrt{2}} \left(\begin{matrix}{- {\rm{i}} } &{ 1} \\ { 1 } &{ - {\rm{i}} } \end{matrix}\right). $$ | (47) |
It can be verified that
$$ {\boldsymbol{N}}_2 := {\boldsymbol{R}}_2 {\boldsymbol{\varLambda}}_2 {\boldsymbol{R}}_2^\dagger = \left(\begin{matrix} { \lambda_{ {\rm{c}} {\rm{r}}} } &{ \lambda_{ {\rm{c}} {\rm{i}}}} \\{-\lambda_{ {\rm{c}} {\rm{i}}} } &{ \lambda_{ {\rm{c}} {\rm{r}}} } \end{matrix}\right), $$ | (48) |
which is real. Of course, to transform
$$ {\boldsymbol{Q}}_2 = \left(\begin{matrix}{\cos\,\theta } &{ \sin\,\theta }\\{-\sin\,\theta } &{ \cos\,\theta}\end{matrix}\right), $$ | (49) |
$$ {\boldsymbol{R}} := \left(\begin{matrix}{{\boldsymbol{R}}_2 } &{ {\bf{0}} }\\{{\bf{0}} } &{ 1}\end{matrix}\right),\; {\boldsymbol{Q}} := \left(\begin{matrix} { {\boldsymbol{Q}}_2 } &{ {\bf{0}}} \\ { {\bf{0}} } &{ 1 } \end{matrix}\right) \;,$$ | (50) |
and
$$ \tilde{\tilde{\tilde{{\boldsymbol{A}}}}} := ( {\boldsymbol{Q}} {\boldsymbol{R}})\tilde{\tilde{{\boldsymbol{A}}}}( {\boldsymbol{Q}} {\boldsymbol{R}})^\dagger = \left(\begin{matrix} { {\boldsymbol{M}} } &{ {\boldsymbol{d}}^\dagger} \\ { {\bf{0}} } &{ \lambda_{ {\rm{r}}} } \end{matrix}\right), $$ | (51) |
where
$$ {\boldsymbol{M}} := \left(\begin{matrix} {\lambda_{ {\rm{c}} {\rm{r}}} - \dfrac{ {\rm{i}}}{2}a_{12}{\rm{e}}^{ {\rm{i}}(2\theta+\phi_{13}-\phi_{23})} }&{ \lambda_{ {\rm{c}} {\rm{i}}} + \dfrac{1}{2}a_{12}{\rm{e}}^{ {\rm{i}}(2\theta+\phi_{13}-\phi_{23})}} \\{-\lambda_{ {\rm{c}} {\rm{i}}} + \dfrac{1}{2}a_{12}{\rm{e}}^{ {\rm{i}}(2\theta+\phi_{13}-\phi_{23})}} & {\lambda_{ {\rm{c}} {\rm{r}}} + \dfrac{ {\rm{i}}}{2}a_{12}{\rm{e}}^{ {\rm{i}}(2\theta+\phi_{13}-\phi_{23})} } \end{matrix}\right) $$ | (52) |
and
$$ {\boldsymbol{d}} := \left({ \frac{ {\rm{i}}}{\sqrt{2}}a_{13}{\rm{e}}^{- {\rm{i}}(\theta+\phi_{13})} + \frac{1}{\sqrt{2}}a_{23}{\rm{e}}^{ {\rm{i}}(\theta-\phi_{23})}, \frac{1}{\sqrt{2}}a_{13}{\rm{e}}^{- {\rm{i}}(\theta+\phi_{13})} + \frac{ {\rm{i}}}{\sqrt{2}}a_{23}{\rm{e}}^{ {\rm{i}}(\theta-\phi_{23})}}\right). $$ | (53) |
That’s all we can do. Now we need to determine
$$ {\boldsymbol{L}}_2 := \left(\begin{matrix} { - {\rm{i}} }& {1} \\ { 1 }& { {\rm{i}}} \end{matrix}\right). $$ | (54) |
However, whatever
The major contributions of this article are:
1) The general complex and real Schur forms are derived constructively, with special consideration for their non-uniqueness. Conditions of uniqueness are found to obtain standardized Schur forms.
2) Conditions for normality and nilpotency are presented. Based on the general complex Schur form, a complex NND is constructed. Based on a special real Schur form, several real NND as well as normal-nonnormal decompositions are constructed. A comparison of advantages and disadvantages between complex and real NNDs is presented.
3) Several confusing points are explained, including the distinction between NND and the TDM proposed by Kolář[15-16], the intrinsic gap between complex and real NNDs.
We hope that the objective of the article, i.e., clarifying the confusion about Schur forms and NNDs existing in research, has been achieved.
Now the open question is, provided so many versions of decompositions, which is the right one to use? Typically, there are two widely adopted usages:
1) Use complex Schur form and complex NND throughout.
2) Use the special block real Schur form eq.(27) and correspongding NND when
But now one has an additional option: use the special block real Schur form eq.(27) and the corresponding NND throughout, i.e., for both real and complex eigenvalue cases. Of course, the above description didn’t take into account the different variants of real NNDs as proposed in section 3, which provides even more options. These options will serve different needs.
The author would like to thank professors WU Jiezhi and YAO Jie for inspiring communication. But the author is solely responsible for the content and point of view of this communication.
[1] |
LI Z. Theoretical study on the definition of vortex[D]. Beijing: Tsinghua University, 2010.
|
[2] |
LI Z, ZHANG X, HE F. Evaluation of vortex criteria by virtue of the quadruple decomposition of velocity gradient tensor[J]. Acta Physica Sinica, 2014, 63(5): 054704. doi: 10.7498/aps.63.054704
|
[3] |
MURNAGHAN F D, WINTNER A. A canonical form for real matrices under orthogonal transformations[J]. Proceedings of the National Academy of Sciences, 1931, 17(7): 417-420. doi: 10.1073/pnas.17.7.417
|
[4] |
SCHUR I. Über die charakteristischen wurzeln einer linearen substitution mit einer anwendung auf die theorie der integralgleichungen[J]. Mathematische Annalen, 1909, 66(4): 488-510. doi: 10.1007/BF01450045
|
[5] |
STICKELBERGER L. Ueber Reelle Orthogonale Substitutionen[M]. Bern: Schweizerische Nationalbibliothek, 1877.
|
[6] |
LÉON A M. Sur l’ Hermitien[J]. Rendicontidel Circolo Matematico di Palermo (1884—1940)
|
[7] |
KEYLOCK C J. Synthetic velocity gradient tensors and the identification of statistically significant aspects of the structure of turbulence[J]. Physical Review Fluids, 2017, 2(8): 084607. doi: 10.1103/PhysRevFluids.2.084607
|
[8] |
KEYLOCK C J. The Schur decomposition of the velocity gradient tensor for turbulent flows[J]. Journal of Fluid Mechanics, 2018, 848: 876-905. doi: 10.1017/jfm.2018.344
|
[9] |
DAS R, GIRIMAJI S S. Revisiting turbulence small-scale behavior using velocity gradient triple decomposition[J]. New Journal of Physics, 2020, 22(6): 063015. doi: 10.1088/1367-2630/ab8ab2
|
[10] |
HOFFMAN J. Energy stability analysis of turbulent incompressible flowbased on the triple decomposition of the velocity gradient tensor[J]. Physics of Fluids, 2021, 33(8): 081707. doi: 10.1063/5.0060584
|
[11] |
ZHU J Z. Thermodynamic and vortic structures of real Schur flows[J]. Journal of Mathematical Physics, 2021, 62(8): 083101. doi: 10.1063/5.0052296
|
[12] |
ZHU J Z. Compressible helical turbulence: fastened-structure geometry and statistics[J]. Physics of Plasmas, 2021, 28(3): 032302. doi: 10.1063/5.0031108
|
[13] |
KRONBORG J, SVELANDER F, ERIKSSON-LIDBRINK S, et al. Computational analysis of flow structures in turbulent ventricular blood flow associated with mitral valve intervention[J]. Frontiers in Physiology, 2022, 13: 806534. doi: 10.3389/fphys.2022.806534
|
[14] |
ARUN R, COLONIUS T. Velocity gradient analysis of a head-on vortexring collision[J]. Journal of Fluid Mechanics, 2024, 982: A16. doi: 10.1017/jfm.2024.90
|
[15] |
KOLÁŘ V. 2D Velocity-field analysis using triple decomposition of motion[C]//Proceedings of the Fifteenth Australasian Fluid Mechanics Conference. Sydney, Australia, 2004.
|
[16] |
KOLÁŘ V. Vortex identification: new requirements and limitations[J]. International Journal of Heat and Fluid Flow, 2007, 28(4): 638-652. doi: 10.1016/j.ijheatfluidflow.2007.03.004
|
[17] |
KRONBORG J, HOFFMAN J. The triple decomposition of the velocity gradient tensor as a standardized real Schur form[J]. Physics of Fluids, 2023, 35: 031703. doi: 10.1063/5.0138180
|
[18] |
ZOU W, XU X Y, TANG C X. Spiral streamline pattern around a critical point: its dual directivity and effective characterization by right eigen representation[J]. Physics of Fluids, 2021, 33(6): 067102. doi: 10.1063/5.0050555
|