Principles and methods of algebraic computations
3.3 The Shifting operation for real matrices
The Shifting operation is a special algebraic transformation which is not very common in the literature of algebra. In Definition 3.3 the Shifting operation was defined for real vectors as the permutation of the leading consecutive zeros of a vector at the end of the vector. Specifically, having a real vector
at = [0, . . . , 0
| {z }
kelements
, ak+1, . . . , an]∈ Rn, ak+1 6= 0
the Shifting operation is defined as
shf : shf(at) = [ak+1, . . . , an, 0, . . . , 0]∈ Rn
and we will simply refer to it as vector Shifting. Naturally, the definition of the Shifting operation can be extended to the case of real matrices.
Definition 3.7. Given a matrix A = [at1, at2, . . . , atn]t∈ Rm×n, the Shifting opera-tion for matrices is defined as the applicaopera-tion of vector Shifting to every row of A.
This transformation will be referred to as matrix Shifting and the shifted form of A will be denoted by
shf(A) , A∗ = [shf(at1), shf(at2), . . . , shf(atn)]t∈ Rm×n
It is important to notice that the Shifting operation, as defined here, permutes the elements of a vector without changing their values and this is a basic require-ment for the Shifting operation in our present study. Regarding the algebraic representation, the vector Shifting can be represented by the multiplication:
shf(at) = at· Jk,n
where Jk,n is an appropriate n× n permutation matrix which is actually a square binary matrix that has exactly one entry 1 in each row and each column and zeros
elsewhere. Each such matrix represents a specific permutation of k elements and for the vector Shifting it has the form:
Jk,n =
where Ii denotes the i× i identity matrix and Oi denotes the i× i zero matrix for i = k, n− k.
Although it is rather simple to represent the vector Shifting with a simple vector-matrix multiplication, it is not obvious how to represent the matrix Shifting transformation, because in general the application of vector Shifting to the rows of a matrix alters the structure of the columns in a non uniform way. The problem of representing the matrix Shifting by using an appropriate matrix-matrix multiplication is very challenging, especially when the modification of the original data is undesired. For the purposes of our study relating to the ERES method, we will investigate the problem of finding an algebraic relation between a real matrix and its shifted form in the class of upper trapezoidal matrices.
Upper trapezoidal matrices occur after the application of Gaussian elimination or other triangularisation methods and they have the following generic form:
A =
Then, the shifted form of A, which is obtained by the matrix Shifting transforma-tion as defined in Definitransforma-tion 3.7, is
A∗ =
In order to simplify the problem, we will focus on finding an algebraic relation between a 2×k matrix, with k > 2, and its shifted form. The proposed constructive method in the following proposition underlies the algebraic representation of matrix Shifting.
Proposition 3.2. If U ∈ R2×k, k > 2, is an upper trapezoidal matrix with rank ρ(U ) = 2 and U∗ ∈ R2×k is the matrix obtained from U by applying the Shifting operation to its rows, then there exists a matrix S∈ Rk×k such that
U∗ = U · S (3.8)
Proof. Let
We can construct the matrix S by following the process:
1. Construct the matrix
H = [Ik|J] ∈ Rk×2k (3.11)
where Ik ∈ Rk×k is the kth identity matrix and J is a permutation matrix of the form:
which permutes the columns of the input matrix U and gives the proper shifting to the 2nd row of U .
2. Multiply the matrices H and U as follows:
U(1) = U· H = (3.13)
Hence, the matrix U(1) has the form:
U(1) = the shifted matrix U∗. The next step is to find a way to extract those two vectors.
3. Denote by
the first 2×2 submatrix of U. Since it is assumed that the diagonal elements of U are uii6= 0, the submatrix U12 is invertible with inverse matrix:
U12−1 =
and hence, the matrix U is right invertible. The right inverse of U can be the matrix:
where O ∈ R(k−2)×2 is a matrix with zero elements.
4. Expand the previously defined matrix H such that H =e h
5. Add proper multiples of the last two columns to all the other columns of U(2) and eliminate the unnecessary entries. For this task, define the matrices:
U =b
Then,
U∗ = U(2)· eV = U · eH· eV (3.21) and therefore, the shifting matrix is given by
S = eH· eV (3.22)
REMARK 3.2. a) The above constructive method requires the original matrix to have full rank. However, as it is obvious in step 3, the selection of the inverse matrix eU is not unique. The main goal here is to create the 2× 2 identity matrix, and for this task we can take at least k− 1 different pairs of columns of U to form its inverse. Furthermore, we can replace the matrix O in eU with any other randomly selected (k− 2) × 2 matrix. Evidently, these changes provide different shifting matrices which transform U into its shifted form U∗. Therefore, the shifting matrix S is not unique.
b) The result in Proposition 3.2 can also be applied to 2× k matrices with more than one consecutive zeros at the beginning of the second row of the given matrix, if we choose the matrices H and eU appropriately.
c) If the shifted matrix U∗ has full rank, the result of Proposition 3.2 can also be applied to itself and then the process is reversed. The shifting matrix S is right invertible and it holds:
U = U∗· S−1
Example 3.1. In this example, we shall demonstrate the steps for constructing the shifted form of the matrix:
U =
"
1 2 3 0 4 5
#
which is the matrix :
U∗ =
"
1 2 3 4 5 0
#
We will follow the next steps, according to the proof of Proposition 3.2 : 1.
H =
1 0 0 0 0 1
0 1 0 1 0 0
0 0 1 0 1 0
∈ R3×6
2.
The matrix S is the shifting matrix and it really holds:
U · S =
However, we could have the same result, if we had used the matrix :
S′ =
where we have taken into account the pseudo-inverse matrix [18] of U .
◮ Matrix Shifting for full rank upper trapezoidal matrices
The process that we used in Proposition 3.2 can be extended in the case of an upper trapezoidal matrix A∈ Rm×n with m≤ n and aii 6= 0 for all i = 1, 2, . . . , m in the form (3.6).
Definition 3.8. a) If the matrix A has the form:
A = every Ji gives the appropriate shifting to each Ai respectively. Therefore,
shf(A) = Xm
i=1
AiJi (3.26)
Since a116= 0, we note that J1 = In, where In is the n× n identity matrix.
If A has full rank, then, since it is defined as an upper trapezoidal with aii 6= 0 for all i = 1, . . . , m, it is right invertible. Let us denote its right inverse by A−1r ∈ Rn×m. The following theorem establishes the connection between a nonsingular upper trapezoidal matrix and its shifted form.
Theorem 3.4. If A ∈ Rm×n, 2 ≤ m < n, is a non-singular upper trapezoidal matrix with rank ρ(A) = m and shf(A)∈ Rm×n is the matrix obtained from A by applying Shifting to its rows, then there exists a square matrix S ∈ Rn×n such that
shf(A) = A· S
The matrix S will be referred to as the shifting matrix of A.
Proof. Let A∗ = shf(A). We shall use the notation described in Definition 3.8 and we will follow the next method to determine the shifting matrix S ∈ Rn×n.
1. Apply to the original matrix A the block matrix S(1)=h
J1 . . . Jm A−1r i
∈ Rn×n(m+1) (3.27)
such that
A(1) = A· S(1) 2. Multiply the matrix A(1) by the block matrix
S(2) = 3. Multiply the matrix A(2) by the block matrix
S(3) =
In the proof of Theorem 3.4 the right inverse matrix A−1r of A is not unique when m < n. Conversely, the pseudo-inverse matrix A† ∈ Rn×m of A can be uniquely determined by calculating the singular value decomposition of A [27], such that
A A†= Im
Therefore, an alternative expression of the previous representation (3.30) of the shifting matrix S can be given, if we use the pseudo-inverse matrix of A. This is
S = Xm
i=1
In− A†A + A†Ai
Ji (3.31)
The expression (3.31) is more appropriate for the numerical computation of the shifting matrix S.
Example 3.2. Consider the following randomly selected matrix:
A =
According to (3.30), the corresponding shifting matrix is:
S1 =
but according to (3.31), the shifting matrix is:
S2 =
1722612 −144305574204 115706143551 459329430653 709807861306
728666
430653 −198896143551 14355112239 231368430653 101936430653
1200059
861306 −418847287102 14355171221 277816430653 −108584430653
55772
143551 −14355192417 14355140064 158752143551 −14355162048
−28710287779 194685287102 14355120032 14355179376 −14355131024
Both S1 and S2 shift the rows of A properly, but they are very different. The computation of their Frobenius norm [18] shows that kS1kF = 24.1376 and kS2kF = 4.0267. Therefore, it seems that the matrix S2, which is computed by using the concept of the pseudo-inverse matrix, is more well-balanced.
If A is a real upper trapezoidal matrix and A∗ denotes its shifted form, then Theorem 3.4 is also applicable to the shifted matrix A∗, provided that A∗ has full rank. However, the Shifting is an operation that alters the structure of a matrix. Therefore, even if the original matrix has full rank, the corresponding shifted matrix may not have full rank. An example is:
A =
In the case where both A and its shifted form A∗ have full rank we obtain the following result. where S−1 denotes the inverse of S.
The previous corollary can be easily proven by following the same steps as in the proof of Theorem 3.4. We only have to
i) change appropriately the set of permutation matrices Ji, i = 1, 2, . . . , m to achieve the proper shifting, and
ii) compute the inverse or pseudo-inverse of A∗.
Therefore, we conclude that the matrix Shifting of a nonsingular upper trapezoidal matrix is a reversible process, unless the shifted matrix is rank deficient.
REMARK 3.3. a) The shifted matrix A∗has full rank if and only if the shifting matrix S has full rank.
b) In general, for two matrices A, B ∈ Rm×n, m≤ n with ρ(A) = ρ(B) = m, we have that
shf(A + B)6= shf(A) + shf(B) shf(A· B) 6= shf(A) · shf(B)
So far, we have analysed the matrix shifting transformation for nonsingular upper trapezoidal matrices and we have established for it a simple algebraic representation by using a matrix-matrix multiplication. This representation has a
key role in the overall algebraic representation of the ERES method, since in every iteration of the main procedure there is always a nonsingular upper trapezoidal matrix, which is formed after the application of the ERE operations.