T-Jordan Canonical Form and T-Drazin Inverse based on the T-Product
In this paper, we introduce the tensor similar transforation and propose the T-Jordan canonical form and its properties. The concept of T-minimal polynomial and T-characteristic polynomial are raised. As a special case, we present properties when two tensors commutes via the tensor T-product. The Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the tensor decomposition theory. T-polar, T-LU, T-QR and T-Schur decomposition of tensors are obtained. When a F-square tensor is not invertible via the T-product, we give the T-group inverse and T-Drazin inverse which can be viewed as the extension of matrix cases. The expression of T-group and T-Drazin inverse are given by the T-Jordan canonical form. The polynomial form of T-Drazin inverse is also proposed. As an application we raise the T-linear system and get its solution. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverse can be given by a limitation process.
Keywords. T-Jordan canonical form, T-function, T-index, tensor decomposition, T-Drazin inverse, T-group inverse, T-core-nilpotent decomposition.
AMS Subject Classifications. 15A48, 15A69, 65F10, 65H10, 65N22.
In linear algebra, the Jordan canonical form of a square matrix has wide applications in many fields . It gives a classification of matrices based on the similar relation which is the main equivalent relation in the matrix theory. This kind of equivalence theory can not be extended to tensors because the multiplication between tensors are not well-defined. The Jordan canonical form is always used to define the matrix functions and give the expressions of Drazin inverse when a matrix is square but not invertible. The group inverse [4, 10] and the Drazin inverse [15, 63] are the two typical kinds of generalized inverses people usually use and research in matrix theories and associative ring theories. In 1999, the Drazin inverse was even extended to -algebras by Koliha [36, 37]. The Drazin inverse is proved to be useful in Markov chain, matrix differential equations and singular linear systems .
There are two important kinds of products between tensors, which is the tensor Einstein product and the tensor T-product. Both the tensor Moore-Penrose inverse and Drazin inverse have been established via Einstein product by Ji and Wei , Jin, Bai, Bentez and Liu  and Sun, Zheng, Bu and Wei  respectively. Recently, the Bernstein concentration inequalities has also been proposed for tensors via the Einstein product by Luo, Qi and Toint . The outer inverse by Einstein product is developed by Stanimirovic et al. 
The tensor T-product introduced by Kilmer  has been proved to be of great use in many areas, such as image processing [33, 34, 49, 55, 59, 70], computer vision [20, 64], signal processing [12, 41, 44, 58], low rank tensor recovery and robust tensor PCA [39, 41], data completion and denoising [14, 28, 29, 42, 44, 47, 48, 53, 60, 65, 66, 67, 68, 69]. An approach of linearization is provided by the T-product to transfer tensor multiplication to matrix multiplication by the discrete Fourier transformation and the theories of block circulant matrices [11, 32]. Due to the importance of the tensor T-product, Lund  gave the definition of tensor functions based on the T-product of third-order F-square tensors in 2018. The definition of T-function is given by
where ‘’ is the block circulant matrix  defined by the F-square tensor . The T-function is also proved to be useful in stable tensor neural networks for rapid deep learning . Special kinds of T-function such as tensor power has been used by Gleich  in Arnoldi methods to compute the eigenvalues of tensors and diagonal tensor canonical form was also proposed.
In this paper, we dedicate to research the F-square tensors and its properties. The organization is as follows. First, the tensor T-similar relationship and the T-Jordan canonical form based on the T-product and their properties are introduced. We find the tensor T-Jordan canonical form is an upper-bidiagonal F-diagonal tensor whose off-F-diagonal entries are the linear combinations of and with coefficients , where and . Based on the T-Jordan canonical form, we give the definition of F-Square tensor polynomial and tensor power series. The convergence radius of tensor series are also given. As an example, we present several power series of classical tensor T-functions and their convergence radius. Then we propose the T-minimal polynomial and T-characteristic polynomial which can be viewed as a special kind of standard tensor function. The Cayley-Hamilton theorem also holds for tensors. As a special case, we discuss the properties when two tensors commutes via the tensor T-product. That is, when two tensors commutes, they can be F-diagonalized by the same invertible tensor. By induction, a family of commutative tensors also hold this property. We also find that normal tensors can be F-diagonalized by unitary tensors. Then we focus on the tensor decomposition. We give the definition of T-positive definite tensors. Then T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained via the T-product.
Unfortunately, when we choose the scalar function to be the inverse function, the induced tensor T-function will not be well defined to invertible tensors. So in the second part of the main results, we discuss the generalized inverse when a F-square tensor is not invertible via the T-product. The T-group inverse and T-Drazin inverse which can be viewed as the extension of matrix cases, including their properties and constructions. We first give the definition of tensor T-index and T-rank. The relationship between the tensor T-index and the tensor T-minimal polynomial is obtained. The existence and uniqueness of T-group inverse is proposed. We also give the condition when the T-group inverse and T-Moore-Penrose inverse are the same. The expression of T-group inverse according to the T-Jordan canonical form is also proposed. Then we focus on the T-Drazin inverse which can be viewed as the generalization of T-group inverse when the T-index of a tensor is known. Based on the T-Jordan canonical form, we obtain the expression of T-Drazin inverse. We find the T-Drazin inverse preserves similarity transformation between tensors. The same as the T-Moore-Penrose inverse , the T-Drazin inverse can also given by the polynomial of the tensor. As an application, we raise the T-linear system when the tensor is singular and give the unique solution according to the T-Drazin inverse. In the last part, we give the T-core-nilpotent decomposition of tensors and show that the tensor T-index and T-Drazin inverse can be given by a limitation process.
2.1 Notation and index
A new concept is proposed for multiplying third-order tensors, based on viewing a tensor as a stack of frontal slices. Suppose two tensors and and denote their frontal faces respectively as and , . We also introduce the operations , and as [20, 34, 35],
and . We can also define the corresponding inverse operation such that .
2.2 The tensor T-Product
(T-product) Let and be two real tensors. Then the T-product is a real tensor defined by
We introduce definitions of transpose, identity and orthogonal of tensors as follows.
(Transpose and conjugate transpose) If is a third order tensor of size , then the transpose is obtained by transposing each of the frontal slices and then reversing the order of transposed frontal slices through . The conjugate transpose is obtained by conjugate transposing each of the frontal slices and then reversing the order of transposed frontal slices through .
(Identity tensor) The identity tensor is the tensor whose first frontal slice is the identity matrix, and whose other frontal slices are all zeros.
It is easy to check that for .
(Orthogonal and unitary tensor) An real-valued tensor is orthogonal if . An complex-valued tensor is unitary if .
For a frontal square tensor of size , it has inverse tensor , provided that
It should be noticed that invertible third order tensors of size forms a group, since the invertibility of tensor is equivalent to the invertibility of the matrix , and the set of invertible matrices forms a group. Also, the orthogonal tensors via the tensor T-product also forms a group, since is an orthogonal matrix.
The concept of T-range space, T-null space, tensor norm, and T-Moore-Penrose inverse is given as follows .
Let be an real-valued tensor.
(1) The T-range space of , , ‘Ran’ means the range space,
(2) The T-null space of , , ‘Null’ represents the null space,
(3) The tensor norm ,
(4) The Moore-Penrose inverse .
2.3 Tensor T-Function
Let be a matrix with spectrum , where and are distinct. Each Jordan block of an eigenvalue has the form
Suppose that has the Jordan canonical form
with blocks of sizes such that , and the eigenvalues .
(Matrix function) Suppose has the Jordan canonical form and the matrix function is defined as
where , and
There are various matrix function properties throughout the theorems of matrix analysis. Here we make review of these properties and the proofs could be found in the excellent monograph .
Suppose A is a complex matrix of size and is a function defined on the spectrum of . Then we have
for all .
By using the concept of T-product, the matrix function is generalized to tensors of size . Suppose we have tensors and , then the tensor T-function of is defined by 
here , where is the vector of all zeros except for the th entry and is the identity matrix, ‘’ is the matrix Kronecker product .
There is another way to express :
Note that on the right-hand side of the equation is merely the matrix function defined above, so the T-function is well-defined.
From this definition, we could see that for a tensor , is a block circulant matrix of size . The frontal faces of are the block entries of , then , where .
In order to get further properties of generalized tensor functions, we make some review of the results on block circulant matrices and the tensor T-product.
 Suppose are block circulant matrices with blocks. Let be scalars. Then , , , , and are also block circulant.
Suppose we have tensors and . Then
(4) , for all ,
(6) , .
3 Main Results
3.1 T-Jordan canonical form
It is a well-known result that every matrix has its Jordan canonical form. The Jordan canonical form is named after Camille Jordan, who first introduced the Jordan decomposition theorem in 1870 and it is of great importance in differential equations, interpolation theory, operation theory, functional analysis, matrix computations and so on [19, 26].
For third order tensors, we can also introduce the Jordan canonical form via the tensor T-product, in this subsection we will define the tensor T-Jordan canonical form. Similar to matrix case, we will only consider complex F-square third order tensors, that is, tensors in the space . Analogous to the linear algebra, we first introduce the following key definition.
(Similar transformation via T-product) Suppose , are two F-square complex tensors. We say is similar to if there exists a invertible tensor satisfies:
Now we dedicate to find the canonical form under the similar relation. We have the following important lemma:
Suppose are complex matrices satisfying
Then the matrices are diagonal (sub-diagonal, upper-triangular, lower-triangular) if and only if the matrices are diagonal (sub-diagonal, upper-triangular, lower-triangular).
By the definition of the Fourier matrix and matrix multiplication each subblock are the linear combination in the complex field of the subblocks .
Furthermore, we reveal the relation between complex matrices and :
where is the a primitive th root of unity and is usually called the phase term.
On the other hand, since the matrix
is the Vandermonde matrix , which is invertible, matrices are also the linear combination of . ∎
(T-Jordan Canonical Form) Let be a complex tensor, then there exists an invertible tensor and a F-upper-bidiagonal tensor such that
From Lemma , for the complex tensor , we have
Since each matrix , () has its Jordan canonical form , , , () so it turns out that
So we have
which is equivalent to
We call the diagonal elements of T-eigenvalues of .
Furthermore, if we suppose
where , () is the -th frontal slice of the tensor . By Lemma , we obtain each is a complex upper bidiagonal matrix and are the linear combination with phase terms of :
Moreover, since the above equations and operations are all invertible, the Jordan canonical form of the tensor is unique. ∎
Now we have established the T-Jordan canonical form, it should be noticed that even though a tensor is a real tensor, its Jordan canonical form could be a complex tensor due to the discrete Fourier transformation.
Recently, Lund  defined the standard tensor T-function as
It has an equivalent definition:
If the tensor has the factorization:
then the above definition of standard tensor T-function is equivalent to
By the definition of tensor T-product, we have
Taking , we have
by comparing with the definition of tensor T-function , we have
which comes to the result. ∎
The standard tensor function holds the following properties:
 Let , and let be defined on a region in the complex plane containing all the spectrum of the matrix . Then the standard tensor function satisfies the following properties:
(1) commutes with ,
(4) If is a polynomial scalar function of degree , . Then the standard tensor function satisfies , where .
Specially, if a tensor has the Jordan canonical form , is a standard tensor function, then
In linear algebra, a nilpotent matrix  is a square matrix satisfies
This concept can be extended to tensors via T-product.
(Nilpotent tensor) A tensor is called nilpotent, if there exists a positive integer such that . If is the smallest number satisfies the equation , then we call is the nilpotent index of .
By the definition of T-eigenvalues, we have the following corollary:
If all the T-eigenvalues of equal to , then is a nilpotent tensor.
3.2 F-square tensor power series
As an direct application of T-Jordan canonical form, we consider the F-square tensor power series as an extension of Lund’s results  and Theorem .
(F-square tensor polynomial) Let be a complex tensor. We call
the polynomial of , whose degree is .
(F-square tensor power series) Let be a complex tensor. We call
the power series of .
By the of Lemma we can transfer the properties of the power series of tensor to the properties of series of the power series of its T-Jordan canonical form , including convergence properties and so on. We first introduce the convergence radius of tensor power series via the T-product, then we will introduce some important kind of F-square tensor series such as tensor exponential, logarithm, triangular functions and so on. By Theorem , we directly give the convergence theorem of tensor series as follows:
(Tensor series convergence) Let be a complex tensor. be a complex power series given by
and its convergence radius is denoted by , and is also called the convergence radius of tensor series . Suppose the maximum of the T-eigenvalues of tensor is . Then if
then the tensor power series converges and the T-eigenvalues of is
By using Theorem , we extend some special kinds of scalar power series to tensor power series via the T-product. There are the following functions:
The standard tensor function , , can be defined to all F-square tensors, since the convergence radius of complex valued power series
are infinity. However, the convergence radius of complex valued power series
are , so the tensor series and can only be defined for F-square tensors whose modules of T-eigenvalues are less than .
Moreover, for the above tensor functions, we have the relation
for all complex tensors and
for tensors whose modules of T-eigenvalues are less than .
Let be two complex tensors commute with each other. Then
Let be a complex tensor. Then there exists a complex tensor , such that
when is invertible, the tensor equation
has a solution.
It should be noticed that the solution of tensor equation is not unique. In fact since
so if , then
This theorem also tells us that every third order tensor has its -th root , that is.
Let be a complex tensor, be a non-zero complex number. Then there exists a complex tensor such that
that is the tensor equation
has a solution.
There exists a complex tensor , such that . Let
Then we have . ∎
Also, the -th root of is not unique. In fact, if , then
For invertible tensors, generally, they may not have its -th root.
We have researched the F-square tensor functions and tensor series. As a general case, we can extend these ideas to block tensors. We have the expression of T-function as follows. Denote the block tensor
by , where be complex tensors. By tensor block multiplication via T-product we have
is the block-wise tensor function via T-product. So it comes to
 Let be a complex tensor with T-eigenvalues , , . is a standard tensor function, then the T-eigenvalues of are , , .
3.3 Commutative tensor family
The set of diagonalizable matrices is an important class of matrices in the linear algebra. In this subsection, we talks about a special class of tensor via T-product, that is the F-diagonalizable tensor.
 (F-diagonalizable tensor) We call a tensor , which has the Jordan decomposition , is a F-diagonalizable tensor if its Jordan canonical form is a F-diagonal tensor, i.e., all the frontal slices of are diagonal matrices.
(T-Commutative) Let , be two tensors. We call tensor commutes with if
For matrix cases, we have the following lemma.
 Let , be two diagonal matrices. If commutes with and is diagonalizable by matrix :
where is a diagonal matrix, then can also be diagonalized by matrix , that is
where is also a diagonal matrix.
If tensors are commutative via T-product and both can be F-diagonalized, then they can be F-diagonalized by the same invertible tensor . In order to prove this result, we shall use the above Lemmas and .
(Diagonalizable simultaneously) Let , be two F-diagonalizable tensors. If commutes with and is F-diagonalizable by tensor :
where is a F-diagonal tensor, then can also be diagonalized by tensor , that is
where is also a F-diagonal tensor.
The equation is equivalent to
Since , we have
Since are all diagonal matrices, by Lemma , we have are all diagonal matrices. By the same kind of method,
So we have and are both diagonalizable matrices, and and commutes with each other, i.e., .
By Lemma , we have can also be diagonalized by the matrix . That is
This is equivalent to
where is also a F-diagonal tensor. ∎
By induction, we have the following generalized case:
(Commutative tensor family) Let , ,, be a family of complex tensors commute with each other via T-product. If is F-diagonalizable by tensor :
where is a F-diagonal tensor, then all the tensors , ,, are F-diagonalizable and can also be diagonalized by tensor , that is,
where is a F-diagonal tensor.
We give the following corollaries without proof which is similar to matrix cases.
(1) Let , be two tensors and . If and are both F-diagonalizable tensors, then is also F-diagonalizable.
(2) Let , be two F-diagonalizable tensors. Then there exists a tensor such that and can be both F-diagonalized by if and only if .
Let , be tensors. is F-diagonalizable. Then if and only if is the polynomial of .
There is a very important kind of tensor which can be unitary similar to a F-diagonal tensor.
(Normal tensor) If is a normal tensor, i.e., commutes with , that is
then there exists a unitary tensor such that
where is a F-diagonal tensor.
Among complex tensors, all unitary, Hermitian, and skew-Hermitian tensors are normal. Likewise, among real tensors, all orthogonal, symmetric, and skew-symmetric tensors are normal. By Theorem 6, all of them can be F-diagonalized by unitary tensors.
There are also many useful results for unitary tensors as follows (without proof).
Let be a complex normal tensor. Then we have
(1) is F-diagonalizable by a unitary tensor.
(2) The Frobenius norm of can be computed by the T-eigenvalues of , that is
(3) The Hermitian part an Skew-Hermitian part of commutes.
(4) is the polynomial (of degree ) of .
3.4 T-characteristic and T-minimal polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity relationship and has the eigenvalues as roots. In complex tensor algebra via the tensor T-product, we can also introduce the concept of T-characteristic polynomial as follows:
(T-characteristic polynomial) Let be a complex tensor, if can be Fourier block diagonalized as
then the T-characteristic polynomial is defined as
where ‘’ means the least common multiplier, is the characteristic polynomial of the matrix .
In matrix theories, the Cayley-Hamilton theorem states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation:
which is an important theorem in matrix theories . For the T-characteristic polynomial of tensors, we also have the following important theorem.
(Caylay-Hamilton Theorem) Let be a complex tensor, be the T-characteristic polynomial of . Then satisfies the T-characteristic polynomial , which means
Since is a tensor in , we operate ‘’ on it
the first step is because of Theorem . By the of Lemma , we get . ∎
The T-eigenvalues of are the roots of the T-characteristic polynomial .
Since similar matrices has the same eigenvalues, it comes to the proof. ∎
Similarly, we define the T-minimal polynomial of as follows:
(Minimal polynomial) Let be a complex tensor, if can be Fourier block diagonalized as
then the T-minimal polynomial is defines as
where ‘’ means the least common multiplier, is the minimal polynomial of matrix , .
For the T-minimal polynomial, we have the similar result as follows:
Let be a complex tensor, be the T-minimal polynomial of . Then satisfies the T-minimal polynomial