I. INTRODUCTION

F.G Frobenius [6] introduced symmetric factorization of matrices in 1910. Frobenius finding was that any given matrix $A\in R^{n\times n}$ can be factorized as $A=S_1{S}_2;S_1=S_1^T,S_2=S_2^T;S_1,S_2\in R^{n\times n}$. Frobenius also stated that one among $S_1,S_2$ or both can be nonsingular. Thus Frobenius hints that there can be left and right symmetrizers for $A$. Here for convenience, we choose $S_1$ as nonsingular. Accordingly there exists a matrix $B=S^{-1}$ so that $BA=S_2$ is symmetric. The nonsingular symmetric matrix $B$ is called a symmetrizer of $A$ [1,3,4,6,21]. In the same year, J. Marty [15] introduced symmetrization of linear integral operators. They were the pioneers who worked in the area of symmetrizing matrices. Their work remained dormant for quite some decades till 1950s. Towards the end of 1950s, matrix symmetrization again drew the attention of many researchers. Notably among them, Professor J.L Howland [8] and his team of researchers, E.J Desautels [3], F.J. Farrel [9] and other mathematicians, Olga Taussky and Hans Zassenhaus [22], Marcus and Khan [14] actively pursued this work of symmetrizing $A$. Marcus and Khan [14] studied in a detailed way, a more general equation $XA=A^{T}X$ where $X$ need not be symmetric and discussed its complete solution. These studies contributed to strengthening many theoretical aspects of symmetrizers. It is reported in Dopico and Uhlig [4] that several attempts by many researchers to compute numerically a symmetrizer during 1960s to early half of 1970s landed into instability and were abandoned mid-way. An account of those failed attempts is presented in Uhlig [24]. It was Olga Taussky and Hans Zassenhaus [22] who gave a new definition for the Frobenius equation as $BA=A^{T}B$, or the similarity transformation $BA{B}^{-1}=A^T$. Later Taussky [21] independently developed the necessary and sufficient conditions and associated theories for $A$ to be symmetrized and the role of symmetrizer $B$ in the study of $A$. Due to these contributions, Frobenius equation became known as Taussky's Symmetrizer Equation. Sen and Venkaiah [23] proposed a solution to similarity symmetrization in the context of solving asymmetric control system problems. It is reported in [4] that method proposed in [23] can be used for low dimensional matrices $A$.

Adhikari's [1] work in year 2000 states that symmetrizing associated asymmetric matrices of linear systems will be convenient. We can then extend the application of established analytical tools for symmetric systems to such cases also. Illustrative examples solving dynamic linear systems in this manner are presented in [1]. What Adhikari [1] introduced is termed as equivalence symmetrization. In line with Adhikari, C.Q. Liu [11] introduced an alternative equivalence symmetrization method. Liu also worked on Adhikari's same illustrative examples in [1] for highlighting its easiness and effectiveness.

It is reported in [4] that first successful numerical computation of a symmetrizer was in 2013 by Frank Uhlig [24,25] based on an iterative linear equation solver of Huang and Nong[10]. Recently in 2016, Froilan Dopico and Frank Uhlig [4] successfully came out with more methods of computing symmetrizers using eigen data and linear equations solver.

The motivation behind [1] was that equivalence transformations are the most general class of nonsingular linear transformations achieving more generality and convenience. We here attempt to transform $BA=A^{T}B$, $A,B\in R^{n\times n}$, $B=B^T$, $\det (B(1:k,1:k))\ne 0$, $k=1:n$, $\det \left(A\right)\ne 0$ into a reduced symmetrizer equation as $DM=M^{T}D$; $D,M\in R^{n\times n}$, where $M$ is similar to $A$ and $D$ is a non singular diagonal matrix. Parter and Youngs [17] considered this topic of symmetrizing $A$ by a diagonal matrix $D\in R^{n\times n}$. Their work was to derive the necessary and sufficient conditions on the existence of $D$ symmetrizing $A$ based on probabilistic rules called conservation and reversibility laws. It was based on George Pimbley's [19] initial value problem for the multi-group transport equation. Pimbley addressed the condition on a non-negative $A\in R^{n\times n}$ that leads to the existence of a positive diagonal matrix $D\in R^{n\times n}$ such that $DA{D}^{-1}$ is a nonnegative symmetric matrix.

The advantages and specialties of reduced symmetrizer equation are highlighted. Frobenius equation [6] is extended to this reduced symmetrizer equation to decompose it into more such reduced equations. Based on this, we prove that nonderogatory $M$ can be expressed as a linear sum of $n^2$ linearly independent matrices having diagonal matrix symmetrizers. A numerical algorithm is developed based on these results for computing a symmetrizer of $A$. This is applied iii) $A^T=S^{-1}AS$ with $S=S^T\succ 0$.

iv) A has real characteristic roots and a full set of characteristic vectors.

Taussky's similarity symmetrization of $A$ and conditions are also discussed in [1,11]. Symmetrization of $A$ can also be achieved by other type of transformations such as equivalent [1], $(A+A^T)/2$, $A{A}^T$, $A^{T}A$, elementary symmetrization [20] etc. Adhikari [1] classified the similarity transformation based symmetrization as first kind. He introduced the equivalence transformation based symmetrization and termed it as second kind. This second kind of symmetrization is defined as follows:

v) A matrix $A$ is symmetrizable of the second kind if and only if there exist two nonsingular matrices $L,R$ such that $\tilde{A}=L^{T}AR$ is symmetric.

This is a more general classification. It includes the first kind as a special case, $L^T=R^{-1}=R^{-T}$. Let $A\in R^{n\times n}$ be nonsingular as well as all its leading principal submatrices, $A(1:k,1:k),k=1:n-1$. Recalling from [7], consider elementary reduction, $L^{T}AR=D$, L, R = U, $D\in R^{n\times n}$, L, U are unit upper triangular matrices and D is a nonsingular diagonal matrix. Thus it is of the second kind. In a reverse way, this second kind symmetrization can be expressed in terms of diagonalization of A as well. We have $L^{T}AU=D$. Let $B=L{L}^T$. Then B is symmetric positive definite and nonsingular. Consider the nonsingular matrices $S_1,S_2\in R^{n\times n}$ given by $S_1=LB$ and $S_2=UB$.

Same $B=L{L}^T$ applied in (1) may be used for simultaneously symmetrizing other coefficient matrices of a given linear system. Now from the definition of a symmetrizer in Frobenious [6], Taussky [21] and Desautels [3], we shall transform the symmetrizer equation using elementary reduction of its symmetrizer.

Let $A\in R^{n\times n}$ be nonsingular. Let it be a symmetrizable matrix as discussed in [1,6,20]. Let $B=B^T\in R^{n\times n}$, $\det (B(1:k,1:k))\ne 0$, $k=1:n$ be a symmetrizer of $A$. Recalling from [7], we can decompose $B$ as $LB{L}^T=D$ applying Gauss Elimination (GE) and is unique. Matrices $D,L\in R^{n\times n}$. Here $L$ is a unit lower triangular matrix and $D$ is a nonsingular diagonal matrix.

Consider $LB{L}^T=D$. We shall rewrite (2) as

Let $M=L^{-T}A{L}^T$. Then equation (3) can be represented as

Lemma 1. Given a symmetrizer $B\in R^{n\times n}$, $\det (B(1:k,1:k))$, $k=1:n\ne 0$ of $A\in R^{n\times n}$, the symmetrizer equation $BA=A^{T}B$ can be transformed into $DM=M^{T}D$ by decomposing $B$ as $LB{L}^T=D$, $L$ is unit lower triangular, $D$ is diagonal and $M=L^{-T}A{L}^T$.

Proof. The proof follows from (2), (3) and (4). Uniqueness of matrices $L,D$ as a pair follows as the decomposition $LB{L}^T=D$ is unique [7].

We see that M is similar to A and D is congruent to B. From (4) it becomes clear that the symmetrizer B should be nonsingular. Frobenius [6] states that in $A=S_1{S}_2,S_1^T=S_1,S_2^T=S_2$, either $S_1$ or $S_2$ can be nonsingular so that A can have a left or right symmetrizer. In (2), we consider the case $B=S_1$ as nonsingular for convenience. Equations (3) and (4) prove that symmetrizing A by B is synchronized with symmetrizing M by D. This leads us to,

Lemma 2. Consider $B=B^T\in R^{n\times n}$, $\det (B(1:k,1:k))$, $k=1:n\ne 0$. Let $LB{L}^T=D$, $L$ is unit lower triangular and $D$ diagonal. Then $B$ symmetrizes $A\in R^{n\times n}$ iff $D$ symmetrizes $M=L^{-T}A{L}^T$.

Proof. We have

Thus (5) is a necessary and sufficient condition for similarity symmetry.

Consider any other row-k, column-k of DM. Applying $d_j,j=1:n$ of (7)

From (8), $D_i$ symmetrizes $M,\det \left(D_i\right)\ne 0$. So $D_i$ is a symmetrizer of M. Hence by Lemma 2, $B_i=L^{-1}{D}_i{L}^{-T}$ is a symmetrizer of A.

Corollary 3.1. If M in Lemma 3 has at least one row-i, column-i of non-zero entries, then it has a diagonal matrix symmetrizer $D_i$ as derived in (7). This diagonal matrix symmetrizer $D_i$ of M is embedded in M itself.

Corollary 3.2. As diagonal matrix $D_i$ can be derived from (7) using arbitrary scalar $x=x_i\ne 0$, infinitely many symmetrizers of $A$ exist against $LB{L}^T=D$.

Remark 3.1. This result can be used to test the symmetrizability of a given matrix M by a diagonal matrix $D_i$ as well as to derive $D_i$ using (7). Coupled with Lemma 2, it can also be used to test the symmetrizability of A by B.

Lemma 4. Let $B=B^T\in R^{n\times n}$, $\det (B(1:k,1:k))$, $k=1:n\ne 0$, $LB{L}^T=D$, be a symmetrizer of $A\in R^{n\times n}$, $L$ unit lower triangular and $D$ diagonal. If $M=L^{-T}A{L}^T$ has a column- $i$ and row- $i$ are without zeros, and $D_1,D_2\in R^{n\times n}$ are its two diagonal matrix symmetrizers, then they are linearly dependent.

Proof. Let $D_1=diag(x_1,x_2,\dots ,x_n),D_2=diag(y_1,y_2,\dots ,y_n)$. Then we have

From (9), it follows that $D_1=(x_i/y_i)D_2,(x_i/y_i)\ne 0$. Hence the result.

Corollary 4.1. If M is without zero entries, then the n symmetrizers $D_i$ of M derived from column-i, row-i; i = 1: n using (7) will be linearly dependent.

Corollary 4.3. Let $A,B=B^T\in R^{n\times n}$. Let $B$ be diagonalizable symmetrizer of $A$ so that $B=L^{-1}D{L}^{-T}$ and $BA=A^{T}B$, $L$ a unit lower triangular matrix and $D$ a nonsingular diagonal matrix. Let $D=\sum _{i=1}^n{\alpha }_i{D}_i$ where $D_i,i=1:n$ constitute a basis of subspace of all diagonal matrices of $R^{n\times n}$. Consider $B=\sum _{i=1}^n{\alpha }_i{B}_i$ where $B_i=L^{-1}{D}_i{L}^{-T},i=1:n$. Then $B_i,D_i,i=1:n$ are not symmetrizers of $A$ and $M=L^{-T}A{L}^T$ respectively.

Remark 4.1. From Corollary 4.3, we see that in several occasions of handling $B_i,i=1:n$, we may miss L but if we use Lemma 3, this will not happen.

Remark 4.2. Reduced equation $DM=M^{T}D$ is more convenient to handle than $BA=A^{T}B$ and from equation (5), these are inter-dependent. As the decomposition $LB{L}^T=D$ is unique in Lemma 4, $M=L^{-T}A{L}^T$ has only one linearly independent diagonal matrix symmetrizer $D$. If $M_1,M_2\in R^{n\times n}$ are linearly dependent, so do their diagonal matrix symmetrizers, say, $D_1,D_2$.

We shall extend the Frobenius equation [6] as applicable to diagonal matrix symmetrizers. We have for $D,Z\in R^{n\times n}$, DZD is always symmetric where D is a nonsingular diagonal matrix and Z is any symmetric matrix.

Lemma 5. A nonsingular diagonal matrix $D\in R^{n\times n}$ will be a symmetrizer of $M\in R^{n\times n}$ iff there exists a matrix $Z=Z^T\in R^{n\times n}$ such that $M=ZD$.

Proof. Suppose D is a symmetrizer of M. Then we have

In (10), let $Z=D^{-1}{M}^T$ and the result follows. Then Z will be nonsingular if and only if M is so. In M, if r columns are dependent then in Z, so much rows will be dependent. Conversely, if there is a matrix $Z;Z=Z^T$ such that

Thus from (11), it is confirmed that $D$ is a symmetrizer of $M$.

Corollary 5.1. As $DM=M^{T}D\iff M=ZD$, let $M=(m_{ij})$, $Z=(z_{ij})$; $i,j=1:n$, then $m_{ij}=m_{ji}=0$ iff $z_{ij}=z_{ji}=0$; $i,j=1:n$.

Corollary 5.2. If $A,B=B^T\in R^{n\times n}$, then $B$ will be a symmetrizer of $A$ iff there exists a matrix $S=S^T\in R^{n\times n}$ such that $A=SB$.

Remark 5.1 By Corollary 5.1, in $Z,M\in R^{n\times n}$, zeros can be only in identical, symmetric positions $(i,j),(j,i),1\le i,j\le n$. This result is stochastically termed

Remark 5.3 Adapting the expression of Z in Remark 5.2 into Frobenius equation [6] and treating $S_2=B$ as nonsingular, we have $S_1=(B^{-1}{A}^T+A{B}^{-1})/2$. The singularity of $S_1$ is depending on the singularity of A. Then we have $A=S_{1}B$, $BA=B{S}_{1}B=A^{T}B$. The symmetrizer B has an independent existence and $S_1$ is dependent on $A,A^T$ and B. This is how the Frobenius equation [6] is extended and how the Taussky's and Frobenius equations are linked.

Remark 5.4 We may see how Pimbley's [19] condition is true for non-negative $M$, , $DM{D}^{-1}=DZ$ is non-negative and symmetric. From (11), Pimbley's equation [19] can be true if $Z$ is a non-negative diagonal matrix.

Remark 5.5 From (11), it follows that if Z is nonsingular, then $Z^{-1}$, Z are symmetrizers of M, $M^T$ respectively. From Desautels [3], the set of symmetrizers of M is a linear subspace of $R^{n\times n}$ of dimension n.

Consider the set of those matrices M, symmetrized by D and may be denoted as $D(M)$. We see that cardinality of largest possible set of linearly independent symmetric matrices $Z_i;i=1,2,\dots \in R^{n\times n}$ which can be applied in M = ZD of (11) is $n(n+1)/2$. Hence D will be a symmetrizer of matrices $M_i=Z_{i}D,i=1:n(n+1)/2\in R^{n\times n}$, which are linearly independent. If D symmetrizes $M_1,M_2$, then it also symmetrizes ${\alpha }_1{M}_1+{\alpha }_2{M}_2$ for nonzero scalars ${\alpha }_1,{\alpha }_2$. So ${\alpha }_1{M}_1+{\alpha }_2{M}_2\in D(M)$. Zero matrix also belongs to $D(M)$. Hence we see that $D(M)$ is a linear subspace of $R^{n\times n}$ of dimension $n(n+1)/2$. Additionally $I_n,M,M^{-1}\in D(M)$. Suppose $Z_i;i=1,2,\dots n$ are nonsingular and linearly independent set of n diagonal matrices. Then $Z_{i}D;i=1,2,\dots ,n$ also will be so. This set of diagonal matrices will span the whole subspace of diagonal matrices of $R^{n\times n}$ and is also a subspace of $D(M)$.