Introduction
Diffusion Tensor Imaging
Diffusion Tensor
and
Diffusion Kurtosis Tensor
in
Biomedical Engineering
by
Diffusion Kurtosis Imaging
D-Eigenvalues and . . .
Further Discussion
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L IQUN Q I
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Department of Applied Mathematics
The Hong Kong Polytechnic University
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Introduction
Outline
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
D-Eigenvalues and . . .
Further Discussion
 Introduction
 Diffusion Tensor Imaging
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 Diffusion Kurtosis Imaging
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 D-Eigenvalues and Other Invariants
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1. Introduction
Introduction
This talk was first given for Gene on November 1, 2007, when he visited me,
accompanied by Michael Ng. He enjoyed this talk very much and brought a
copy of this talk. He even asked several references, and I promised to send him
later. It was totally unexpected that he departed only after 15 days. I was totally
shocked by the sad news. I updated this talk, and present it now to memorize
Gene.
Diffusion Kurtosis Imaging
Diffusion Tensor Imaging
One year ago, in February of 2007, at the Chinese New Year time, I received
an e-mail from a Hong Kong University second year biomedical engineering
student. He asked questions about my paper:
[1]. L. Qi, “Eigenvalues of a real supersymmetric tensor”, Journal of Symbolic
Computation 40 (2005) 1302-1324.
This surprised me. Why a second year biomedical engineering student would
be interested in my mathematical paper on eigenvalues of higher order tensors?
Eventually, started from the responses to that e-mail, I established research collaboration with that student’s supervisor, Professor Ed Xuekui Wu, Associate
Professor, Director of Laboratory of Biomedical Imaging and Signal Processing, The University of Hong Kong.
D-Eigenvalues and . . .
Further Discussion
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1.1.
Introduction
Collaboration with Ed X. Wu
Diffusion Tensor Imaging
Since then, we have written four papers together. The first paper in this collaboration has already been accepted for publication:
Diffusion Kurtosis Imaging
[2]. L. Qi, Y. Wang and E.X. Wu, “D-eigenvalues of diffusion kurtosis tensors”,
to appear in: Journal of Computational and Applied Mathematics.
Further Discussion
[3]. L. Qi, D. Han and E.X. Wu, “Principal invariants and inherent parameters
of diffusion kurtosis tensors”, Manuscript, Department of Applied Mathematics,
The Hong Kong Polytechnic University, May 2007.
[4]. D. Han, L. Qi and E.X. Wu, “Extreme diffusion values for non-Gaussian
diffusions”, Manuscript, Department of Applied Mathematics, The Hong Kong
Polytechnic University, August 2007.
[5]. E.X. Wu, E.S. Hui, M.M. Cheung and L. Qi, “Towards better MR characterization of Neural tissues using directional diffusion kurtosis analysis”,
Manuscript, Laboratory of Biomedical Imaging and Signal Processing, The
Hong Kong University, January, 2008.
Here, Matthew M. Cheung is the second year biomedical engineering student,
who sent me e-mails one year ago.
D-Eigenvalues and . . .
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Introduction
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
D-Eigenvalues and . . .
2. Diffusion Tensor Imaging
Further Discussion
Diffusion magnetic resonance imaging (D-MRI) has been developed in biomedical engineering for decades. It measures the apparent diffusivity of water
molecules in human or animal tissues, such as brain and blood, to acquire biological and clinical information. In tissues, such as brain gray matter, where
the measured apparent diffusivity is largely independent of the orientation of the
tissue (i.e., isotropic), it is usually sufficient to characterize the diffusion characteristics with a single (scalar) apparent diffusion coefficient (ADC). However,
in anisotropic media, such as skeletal and cardiac muscle and in white matter,
where the measured diffusivity is known to depend upon the orientation of the
tissue, no single ADC can characterize the orientation-dependent water mobility in these tissues. Because of this, a diffusion tensor model was proposed
years ago to replace the diffusion scalar model. This resulted in diffusion tensor
imaging (DTI).
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2.1.
Diffusion Tensor
Introduction
Diffusion Tensor Imaging
A diffusion tensor D is a second order three dimensional fully symmetric tensor.
Under a Cartesian laboratory co-ordinate system, it is represented by a real three
dimensional symmetric matrix, which has six independent elements D = (dij )
with dij = dji for i, j = 1, 2, 3. There is a relationship
ln[S(b)] = ln[S(0)] −
3
X
bdij xi xj .
(1)
Diffusion Kurtosis Imaging
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i,j=1
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Here S(b) is the signal intensity
P3 at2 the echo time, x = (x1 , x2 , x3 ) is the unit
direction vector, satisfying i=1 xi = 1, the parameter b is given by
δ
b = (γδg)2 (∆ − ),
3
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γ is the proton gyromagnetic ratio, ∆ is the separation time of the two diffusion
gradients, δ is the duration of each gradient lobe. See Figure 1 on the next
page. There are six unknown variables dij in the formula (1). By applying the
magnetic gradients in six or more non-collinear, non-coplaner directions, one
can solve (1) and get the six independent elements dij .
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2.2.
Eigenvalues of Diffusion Tensor
However, such elements dij cannot be directly used for biological or clinical
analysis, as they vary under different laboratory coordinate systems. Thus, after obtaining the values of these six independent elements by MRI techniques,
the biomedical engineering researchers will further calculate some characteristic
quantities of this diffusion tensor D. These characteristic quantities are rotationally invariant, independent from the choice of the laboratory coordinate system.
They include the three eigenvalues λ1 ≥ λ2 ≥ λ3 of D, the mean diffusivity
(MD ), the fractional anisotropy (F A), etc. The largest eigenvalue λ1 describes
the diffusion coefficient in the direction parallel to the fibres in the human tissue. The other two eigenvalues describe the diffusion coefficient in the direction
perpendicular to the fibres in the human tissue. The mean diffusivity is
λ1 + λ 2 + λ 3
MD =
,
3
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while the fractional anisotropy is
r s
3 (λ1 − MD )2 + (λ2 − MD )2 + (λ3 − MD )2
FA =
,
2
λ21 + λ22 + λ23
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where 0 ≤ F A ≤ 1. If F A = 0, the diffusion is isotropic. If F A = 1, the
diffusion is anisotropic.
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2.3.
Diffusion Tensor Imaging
Introduction
The diffusion tensor imaging model (DTI) is now used widely in biological and
clinical research. There are many papers on DTI in biomedical engineering
journals, in particular, MRI journals. Even in the Mainland China, there are a
number of papers on DTI in Chinese.
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
D-Eigenvalues and . . .
Further Discussion
A review paper on the technique of DTI is:
[6]. P.J. Basser and D.K. Jones, “Diffusion-tensor MRI: theory, experimental
design and data analysis - a technical review”, NMR in Biomedicine 15 (2002)
456-467.
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A paper on the DTI technique in Chinese is:
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[7]. D. Li, S. Bao, C. Zhu and L. Ma, “Computing the measures of DTI based
on PC and Matlab”, Chinese Journal of Medical Imaging Technology 20 (2004)
90-94. (in Chinese)
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A figure from their paper is on the next page.
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[8]. W. Chong and J. Zhao, “Role of diffuse tensor imaging”, Chinese Journal
of CT and MRI 13 (2005) 49-51, (in Chinese)
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reported eight brain and spinal diseases detectable by the DTI methods.
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Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
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Introduction
3. Diffusion Kurtosis Imaging
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
However, DTI is known to have a limited capability in resolving multiple fibre
orientations within one voxel. This is mainly because the probability density
function for random spin displacement is non-Gaussian in the confining environment of biological tissues and, thus, the modeling of self-diffusion by a
second order tensor breaks down. Recently, a new MRI model is presented by
[9]. J.H. Jensen, J.A. Helpern, A. Ramani, H. Lu and K. Kaczynski, “Diffusional
kurtosis imaging: The quantification of non-Gaussian water diffusion by means
of magnetic resonance imaging”, Magnetic Resonance in Medicine 53 (2005)
1432-1440;
[10]. H. Lu, J.H. Jensen, A. Ramani and J.A. Helpern, “Three-dimensional characterization of non-Gaussian water diffusion in humans using diffusion kurtosis
imaging”, NMR in Biomedicine 19 (2006) 236-247.
They propose to use a fourth order three dimensional fully symmetric tensor,
called the diffusion kurtosis (DK) tensor, to describe the non-Gaussian behavior. The values of the fifteen independent elements of the DK tensor W can
be obtained by the MRI technique. The diffusion kurtosis imaging (DKI) has
important biological and clinical significance.
D-Eigenvalues and . . .
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Introduction
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
3.1.
Diffusion Kurtosis Tensor
D-Eigenvalues and . . .
A diffusion kurtosis tensor W is a fourth order three dimensional fully symmetric tensor. Under a Cartesian laboratory co-ordinate system, it is represented by
a real fourth order three dimensional fully symmetric array, which has fifteen
independent elements W = (wijkl ) with wijkl being invariant for any permutation of its indices i, j, k, l = 1, 2, 3. The relationship (1) can be further expanded
(adding a second Taylor expansion term on b) to:
ln[S(b)] = ln[S(0)] −
3
X
1
bdij xi xj + b2 MD2
6
i,j=1
3
X
wijkl xi xj xk xl .
(2)
i,j,k,l=1
There are fifteen unknown variables wijkl in the formula (2). By applying the
magnetic gradients in fifteen or more non-collinear, non-coplaner directions,
one can solve (2) and get the fifteen independent elements wijkl .
Further Discussion
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D-Eigenvalues and . . .
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3.2.
Diffusion Kurtosis Imaging
The diffusion kurtosis imaging (DKI) has important biological and clinical significance. The authors of [9] found sharp differences between the diffusion kurtosis in white and gray matters. They believe that DKI is potentially of value for
the assessment of neurologic diseases, such as multiple sclerosis and epilepsy,
with associated white matter abnormalities. Additional, DKI may be useful for
investigating abnormalities in tissues with isotropic structures, such as gray matter, where techniques like diffusion tensor imaging (DTI) are less applicable.
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Introduction
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
D-Eigenvalues and . . .
Further Discussion
4. D-Eigenvalues and Other Invariants
Again, the fifteen elements wijkl vary when the laboratory co-ordinate system is
rotated. What are the coordinate system independent characteristic quantities of
the DK tensor W ? Are there some type of eigenvalues of W , which can play a
role here?
E.X. Wu is a friend of J.H. Jensen, the main author of [9] and [10]. He and
his group studied these questions. They searched by google possible papers on
eigenvalues of higher order tensors. They found my paper [1]. This resulted in
the surprising e-mail to me in February, 2007.
In [2-5], we answered these questions.
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4.1.
The Apparent Kurtosis Coefficient
In [2], we use x = (x1 , x2 , x3 )T to denote the direction vector. Then the apparent
diffusion coefficient (ADC)
Introduction
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
2
Dapp = Dx ≡
3
X
D-Eigenvalues and . . .
dij xi xj .
Further Discussion
i,j=1
A key formula for the DK tensor W is as follows:
Kapp
MD2
= 2 W x4 ,
Dapp
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(3)
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where Kapp is the apparent kurtosis coefficient at the direction x, and
W x4 ≡
3
X
wijkl xi xj xk xl .
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i,j,k,l=1
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Note that D and W are fully symmetric. Furthermore, D is positive definite. It
is easy to see that maximizing (minimizing) Kapp is equivalent to the following
maximization (minimization) problem:
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4
max W x
s.t. Dx2 = 1.
(4)
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Introduction
Diffusion Tensor Imaging
4.2.
D-Eigenvalues of The Diffusion Kurtosis Tensor
Diffusion Kurtosis Imaging
Obviously, the critical points of problem (4) satisfy the following equation for
some λ ∈ <:
W x3 = λDx,
(5)
Dx2 = 1.
A real number λ satisfying (5) with a real vector x is called a D-eigenvalue of
W , and the real vector x is called a D-eigenvector of W associated with the
D-eigenvalue λ.
Theorem 4.1 D-eigenvalues always exist. They are invariant under co-ordinate
rotations. If x is a D-eigenvector associated with a D-eigenvalue λ, then
λ = W x4 .
D-Eigenvalues and . . .
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(6)
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MD2 λmax ,
The largest AKC value is equal to
and the smallest AKC value is equal
2
to MD λmin , where λmax and λmin are the largest and the smallest D-eigenvalues
of W respectively.
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Diffusion Tensor Imaging
4.3.
Diffusion Kurtosis Imaging
Computation of D-Eigenvalues
D-Eigenvalues and . . .
In [2], we presented a direct method to find all the D-eigenvalues and their corresponding D-eigenvectors. The key idea there is to reduce the four variable
polynomial system (5) to a polynomial system of two variables u and v. Then
regard it as a system of two polynomial equations of one variable u, whose coefficients are polynomials of v. It has complex solutions for u if and only if its
resultant vanishes. By the Sylvester theorem, for this problem, its resultant is
equal to the determinant of a 7 × 7 matrix, which is a one-dimensional polynomial of v.
To find the approximate solutions of all the real roots of a one-dimensional polynomial to any given error tolerance, we can use the Sturm Theorem. We then
substitute them back to find the corresponding approximate real solutions of
u. Correspondingly, approximate values of all the D-eigenvalue and their Deigenvectors can be obtained.
Further Discussion
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4.4.
Introduction
A Numerical Example
Diffusion Tensor Imaging
This example is derived from data of MRI experiments on rat spinal cord specimen fixed in formalin. The MRI experiments were conducted on a 7 Tesla MRI
scanner at Laboratory of Biomedical Imaging and Signal Processing at The University of Hong Kong. This example is taken from the white matter.
The six elements of the diffusion tensor D are d11 = 0.1755, d12 =
0.0035, d13 = 0.0132, d22 = 0.1390, d23 = 0.0017, d33 = 0.4006 in unit of
10−3 square mm per second, and the fifteen independent elements of the diffusion kurtosis tensor W are w1111 = 0.4982, w2222 = 0, w3333 = 2.6311,
w1112 = −0.0582, w1113 = −1.1719, w1222 = 0.4880, w2223 = −0.6162,
w1333 = 0.7639, w2333 = 0.7631, w1122 = 0.2236, w1133 = 0.4597, w2233 =
0.1519, w1123 = −0.0171, w1223 = 0.1852 and w1233 = −0.4087, respectively.
It is easy to find that
2
d
+
d
+
d
11
22
33
MD2 =
= 5.6813 × 10−8 .
3
Diffusion Kurtosis Imaging
Using the method provided in [2], we compute all the D-eigenvalues of W , and
the associated D-eigenvectors, which are listed on the next page.
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4.5.
D-eigenvalues and eigenvectors of W
Introduction
(1)
λ1 = 3.8340 × 10−7 , x(1) = (−26.6953, −76.0271, 13.2301)T , Kapp = 2.1782.
−7
λ2 = 2.4323 × 10 , x
(2)
= (8.3561, −69.1354, 28.6108)
(2)
, Kapp
T
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
= 1.3819.
D-Eigenvalues and . . .
(3)
λ3 = 0.6773 × 10−7 , x(3) = (58.9844, −42.6590, 17.9274)T , Kapp = 0.3848.
Further Discussion
(4)
λ4 = 3.8173 × 10−7 , x(4) = (−62.4736, −35.3967, 20.0392)T , Kapp = 2.1687.
(5)
λ5 = 2.4900 × 10−7 , x(5) = (29.1437, −52.2572, 33.9600)T , Kapp = 1.4146.
−7
λ6 = 1.0247 × 10 , x
(6)
= (34.8925, 49.2278, 31.7172)
−7
λ7 = −0.0738 × 10 , x
−0.0419.
(7)
T
(6)
, Kapp
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= 0.5822.
= (−21.9794, −31.4925, 44.7823)
T
(7)
, Kapp
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=
(8)
λ8 = 2.0092 × 10−7 , x(8) = (24.4491, −12.3897, 46.0146)T , Kapp = 1.1415.
−7
λ9 = 2.0563 × 10 , x
(9)
= (−12.2897, 23.6850, 47.6412)
−7
(10)
−7
(11)
λ10 = 5.3545 × 10 , x
(9)
, Kapp
T
= (−66.1780, 11.3946, 25.5877)
T
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= 1.1682.
(10)
, Kapp
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= 3.0420.
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λ11 = 2.2194 × 10 , x
−7
λ12 = −1.2420 × 10 , x
= (11.5201, 18.0765, 47.7258)
(12)
T
= (65.6942, 7.1795, 21.9765)
−7
(11)
, Kapp
= 1.2609.
(12)
, Kapp
= −0.7056.
T
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−7
We see that λmin = −1.2420 × 10 , λmax = 5.3545 × 10 .
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Introduction
Diffusion Tensor Imaging
Diffusion Kurtosis Imaging
4.6.
Principal Invariants of the Diffusion Kurtosis Tensor
In [3], we presented several principal invariants of the diffusion kurtosis tensors,
and their computational formulas. These principal invariants include the largest,
the smallest and the average apparent kurtosis coefficients (AKC) values, which
are invariant from the co-ordinate system choices, and some parameters measured in the inherent co-ordinate system, which is formed by the eigenvector
system of the second order diffusion tensor. We studied their computation formulas and relationships. We made numerical experiments, based upon the magnetic resonance diffusion data acquired out of rat spinal cord samples fixed in
formalin. See the figure on the next page. The MRI experiments were conducted
on a 7 Tesla MRI scanner at Laboratory of Biomedical Imaging and Signal Processing at The University of Hong Kong, led by E.X. Wu.
In [4], we studied properties and formulas of extreme diffusion values of diffusion kurtosis imaging (DKI).
D-Eigenvalues and . . .
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5. Further Discussion
D-Eigenvalues and . . .
Further Discussion
We see that the diffusion tensor and the diffusion kurtosis tensor are important
physical quantities in biomedical engineering. While eigenvalues and invariants
of a second order tensor are well studied, the study on eigenvalues and invariants
of a fourth order tensor has just started. There are still many things unexplored
there. Here, we have the following comments:
1. We plan to put our code for calculating D-eigenvalues, written by Dr. Deren
Han, on my website, such that the biomedical engineering researchers can use
it.
2. There are some other non-Gaussian models, such as q-space and higher order
tensor models in MRI. It will be interesting if we can apply the techniques of
[2-4] to them.
3. We may find more than 15 meaningful invariants of W . Apparently, only 15
of them can be independent. Can we identify some independence or dependence
relationships among them?
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5.1.
Diffusion Tensor Imaging
Discussion on Invariants
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D-Eigenvalues and . . .
We now know more about invariants:
Further Discussion
1. In the literature invariants normally refer to the polynomials in the elements
of a tensor invariant under a specific group. Here we discuss about orthogonal
group.
2. For a symmetric mth order tensor, there is only one independent linear invariant, when m is even, and no linear invariant, when m is odd. For a fourth
order symmetric tensor A, this linear invariant is
X
X
aiiii + 2
aiijj .
i=1,··· ,n
i,j=1,··· ,n,i6=j
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For n = 3, in [3], we show that this invariant is related with the average value
of AKC.
3. A fourth order three dimensional symmetric tensor has 12 independent “basic” invariants. The other invariants can be represented by them.
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5.2.
Number of D-Eigenvalues
Recently, Professor K.C. Chang at Peking University studied eigenvalues of real
symmetric tensors with his co-authors:
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[11]. K.C. Chang, K. Pearson and T. Zhang, “On eigenvalues of real symmetric
tensors”, Preprint, School of Mathematical Science, Peking University, Beijing,
China, February 2008.
In that paper, Professor Chang and his co-authors unified the definitions of
H-eigenvalues, Z-eigenvalues and D-eigenvalues, and show that for a real
n-dimensional mth order symmetric tensor, there are at least n H-/Z-/Deigenvalues.
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