Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns including hyperdominant one is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kind RLC networks and in generation of their equivalent realizations.


Introduction
During the past few decades,many papers 1-16 have studied the inverse eigenvalue problems IEPs of various types.The solution existence of the specific IEPs was generally considered in 1, 3-8, 10, 11, 13, 14 without explicit formulation of the corresponding procedure for solution construction, whereas in 2, 9, 12, 15, 16 this has been accomplished.The main result of 16 is the proof that IEP of symmetric hyperdominant hd matrices with assigned nonnegative spectrum has at least one solution which has also been constructed.This settled an old IEP opened in 17 .Hyperdominant matrices have nonnegative diagonal and nonpositive off-diagonal entries and nonnegative hd margins of rows hd margin of a row is the sum of entries in that row .The tool used in 16 to construct the nth-order hd matrix with assigned spectrum was the nth-order orthogonal Hessenberg matrix constructed as a special product of n − 1 plane rotations 15 .Hessenberg matrices naturally arise in study of symmetric tridiagonal matrices, skew symmetric, and orthogonal matrices 13, 14, 18 .A matrix is upper lower Hessenberg if its entry k, m vanishes whenever k > m 1 m > k 1 .
In practical work, it is commonly assumed to be better not to form Hessenberg matrices explicitly, but to keep them as products of plane rotations.On the other hand, explicit construction of real symmetric matrices with nonnegative spectrum, which either have hd sign pattern or are truly hd, is proved to be an inevitable task in considering the synthesis of driving-point immittance functions of passive, transformerless, common-ground, two element-kind RLC networks and in generation of their equivalent realizations [17][18][19] .RLC networks are comprised solely of resistors R , inductors L , and capacitors C .Drivingpoint immittance function of a lumped, time invariant, linear electrical network is either a driving-point impedance Z s , or a driving-point admittance Y s Z −1 s s σ j•ω is the complex frequency; σ, ω are real numbers; j : √ −1 .It is well known that a real rational function in s can be driving-point immittance function of RLC network if and only if it is positive real function in s; or similarly, a necessary condition for a stable square matrix W s of real rational functions in s to be driving-point immittance matrix of a passive RLC network is that W s be positive real matrix 20, 21 .A few tests for ascertaining positive real properties of functions and/or matrices can be found in 20, 21 .In 22 it has been pointed out the role of hd matrices in synthesis of both passive and active, transformerless, commonground multiports.Unlike 16 , this paper presents explicit construction of entries of real symmetric matrices with arbitrarily assigned spectrum and the entries of the corresponding orthogonal modal matrices.It also presents explicit construction of real symmetric matrices with assigned spectrum and with specific sign patterns including hd one .Thereof, a solution to the IEP of symmetric, truly hd matrices with assigned nonnegative spectrum is produced.Some of the obtained results are then applied in synthesis of driving-point immitances of transformerless, common-ground, two-element-kind RLC networks and in generation of their equivalent realizations.The two proposed realization procedures are illustrated by an example.Throughout the paper ⊕ denotes direct sum, x T denotes transpose of x, bold capital letters denote matrices and I k stands for the kth-order unit or identity matrix.

Explicit solution to the IEP of real symmetric matrices by using canonic orthogonal transformations
Let {λ 1 , λ 2 , . . ., λ n } be assigned spectrum of the sought real symmetric matrices and let G 1 : diag λ 1 , λ 2 , . . ., λ n be n × n spectral matrix.Consider a set of 2 × 2 orthogonal matrices and x 2 : c 1 ε 1 , and let us firstly introduce in 2.5 the following notation:

2.6
Thereafter, observing the partition of G 2 and G 3 obtained in 2.5 it can readily be anticipated the partition of subsequent matrices G k k 4, . . ., n − 2 as follows: where

2.9
For k 2, . . ., n − 3, let us define: Then, from 2.8 -2.9 it follows the identification which enables the partition of G k 1 in 2.9 to be like that of G k in 2.8 , and that partition of x k 1 be rather simple

2.12
Since G n−1 : it follows from 2.12 -2.13 we obtain from 2.14 -2.15 the partition of G n which is amenable to the production of its entries in explicit form and is suitable for further discussion about solving some specific IEPs For k 2, . . ., n, we consecutively obtain from λ * 1 : . ., n − 1 , then from 2.17 it follows that

2.18
Observe that it is not necessary to calculate "ψ"s from 2.18 , but only the modified eigenvalues from 2.17 since it holds As it is x 2 c 1 ε 1 , then for k 2, . . ., n − 2 from 2.10 it follows that

2.22
If we now suppose that then since according to 2.2 , it holds U k 1 :

2.24
By using 2.2 , 2.23 -2.24 , it follows that and thereby it is proved our previous assumption that

2.26
The entries of the orthogonal lower Hessenberg matrix U u km k, m 1, . . ., n are defined as follows:

2.27
By using the similar arguments as in derivation of entries of matrix U, the orthogonal matrix V which is to be produced by using 2.4 can be shown to take on upper Hessenberg form.Proving of this fact goeswith similar paces that were used for obtaining U and it is left to the reader.

The explicit solution of the IEP of real symmetric matrices with some specific sign patterns
Let the real eigenvalues from the spectrum {λ 1 ,λ 2 , . . .,λ n } be arbitrarily enumerated, thereby establishing the sequence {λ k } k 1, . . ., n .The nonnegative sequence will be denoted by {λ k } ≥ 0, and the nonpositive one by {λ k } ≤ 0 k 1, . . ., n .Firstly, we will prove two lemmas.

3.1
From 2.17 and the last of inequalities 3.

3.2
From 2.17 and the last of inequalities 3.
This completes the proof of lemma.For a nonpositive sequence, an analogous lemma can be formulated.

Lemma 3.2. If the sequence {λ
Proof.It is similar to that of Lemma 3.1, but in this case the diagonal entries of G n are nonpositive, that is, λ * n ≤ 0 and ψ mm ≤ 0 m 1, . . ., n − 1 , no matter whether the sequence {λ k } ≤ 0 is increasing or decreasing see 2.18 .Now, we shall formulate a new theorem related to explicit solving of IEP of real symmetric matrices with some specific sign patterns.
thus obviously making the matrix G n 2.21 with sparse structure.By using 2.17 -2.18 , 2.21 and both Lemmas, we can readily infer that if θ k ∈ 0, π/2 k 1, . . ., n − 1 and the sequence {λ m } m 1, . . ., n is strictly monotone, then matrix G n 2.21 is produced with no zero entries in all three considered cases.
Also, when the sequence {λ m } m 1, . . ., n is increasing decreasing , then the sequence {λ −1 m } m 1, . . ., n is decreasing increasing .These facts and Theorem 3.3 offer a possibility of determining the sign pattern of G −1 n without really inverting G n .Furthermore, by using 2.17 1 U T will be nonnegative matrix.Since d m > 0 m 1, . . ., n , then the nonsingular symmetric matrix DG n D is produced with hd sign pattern, but it may not be truly hd, unless hd margin of each of its rows or columns is nonnegative recall that hd margin of a row or a column is sum of all entries in that row or column .If G n g km k, m 1, . . ., n , then hd margin p k of the kth row or the kth column in DG n D is given by 3.6 Let we arbitrarily select α k > 0 k 1, . . ., n and let a : 1 and p Da > 0 n,1 .This not only means that DG n D has hd sign pattern, but that it is truly hd furthermore.Obviously, as much as "α"s are assumed greater, the greater will be row column hd margins of DG n D. This completes the proof of theorem.

Explicit solution of IEP of hd matrices with uncommitted and with assigned nonnegative spectrum
Theorem 4.1.Let θ k k 1, . . ., n − 1 be a set of angles selected from the range 0, π/2 and let {λ 1 , λ 2 , . . ., λ n } be uncommitted nonnegative spectrum of the real symmetric matrix G n UG 1 U T G 1 diag λ 1 , λ 2 , . . ., λ n which is to be produced as truly hd.Suppose that through enumeration of eigenvalues the sequence {λ m } ≥ 0 m 1, . . ., n is made increasing.Then, matrix G n given by 2.21 will be truly hyperdominant if λ 1 is sufficiently great.
Proof.Since by assumption the conditions of Theorem 3.3 Case 1 are satisfied, then G n produced by using 2.21 has hd sign pattern.As it is ε k λ * k − λ k 1 a k k 1, . . ., n − 1 , then from 2.17 -2.18 , 2.21 it follows that hd margin p m of the mth row or column from G n m 1, . . ., n can be in general represented as where "α" coefficients are defined as follows: According to Case 1 of Theorem 3.3, both a k and c k are nonnegative when θ k ∈ 0, π/2 k 1, . . ., n − 1 .Then, from 4.2 we see that α m 1 ≥ 0 m 1, . . ., n , whereas other "α"s may be nonpositive.Since "α"s depend only on selection of "θ"s, then by presuming λ 1 λ 2 • • • λ n λ / 0, we obtain from 2.21 G n λI n and p m λ m 1, . . ., n and from 4.1 we conclude that in general it holds: Although G n is produced with hd sign pattern, it will not be truly hd unless each of its row column hd margins is nonnegative, that is, p The column vector p with entries 4.1 can be written as From 4.4 we finally obtain
Presentation of explicit solution to the IEP of truly hd matrices with assigned nonnegative spectrum is now in order.It has been proved in 16 that this IEP always has at least one solution and that infinitely many others can be produced thereof by using Givens rotations.Solution of that IEP is important in electrical network synthesis of drivingpoint immittance functions and matrices of both passive and active, common-ground, transformerless, two-element-kind RLC networks and in generation of various classes of canonic and noncanonic equivalent realizations 19, 22 .In 16 we have proved the existence of solution to the IEP of hd matrices with assigned nonnegative spectrum, but here we shall present the explicit construction of solution matrix entries by using other arguments.This represents the explicit solution of the problem opened in 17 .
Theorem 4.2.For any set of real nonnegative numbers {λ 1 , λ 2 , . . ., λ n } there always exists at least one (and infinitely many) n × n real symmetric hyperdominant matrices having these numbers as eigenvalues.In other words, IEP of symmetric hd matrices with assigned nonnegative spectrum always has at least one solution.
Proof.We will take the same assumptions as in Theorem 4.1, except for θ k ∈ 0, π/2 k 1, . . ., n − 1 .Through enumeration of eigenvalues, the nonnegative sequence {λ m } m 1, . . ., n is made increasing.Then, according to Theorem 3.3 Case 1 , the symmetric matrix UG 1 U T truly hd, we will prove the existence of such "θ"s that make all "α"s and hence all "p"s in 4.1 nonnegative.Let we introduce the following positive sequence

Mathematical Problems in Engineering
Then, by using 4.2 we obtain a consistent set of inequalities that ensure nonnegativity of all "α"s in 4.1 For p 2 from 4.8 we obtain M 2 ≥ c 1 /a 1 and from 4.7 1/a 1 and p 1 λ 1 .From 4.11 -4.12 it follows that a n−1 c n−1 ↔ θ n−1 π/4 and α n−1 n α n n 0 inequalty 4.12 is the same as 4.9 if k n − 1 M n 1 .For k 2, . . ., n − 2 , we obtain from 4.9 M k 1 ≤ c k /a k and for q 3, . . ., n − 1 , we obtain from 4.10 M q ≥ c q−1 /a q−1 .To summarize, we have proved that: a M r c r−1 /a r−1 , for r 2, . . ., n − 1 and b {α And finally, from 4.10 we obtain α n q 0 for q 2, . . ., n, α n 1 1 and p n λ 1 .Since the matrix G n has hd sign pattern and each of its row column hd margins is equal to λ 1 ≥ 0, then G n is truly hd matrix.This completes the proof of the theorem.

4.13
By using 2.17 -2.18 , 2.21 , 4.13 we can easily calculate all entries of the initial hd matrix G n .Other hd matrices having the same spectrum can be produced thereof by application of Givens rotations, one at a time.

Application of the obtained results in electrical network synthesis
It is well known that synthesis methods of passive, common-ground, transformerless, twoelement-kind RLC networks yield topological configurations which are severely restricted by the method chosen 19 .By using of the results above, a new class of non-canonic, drivingpoint immittance realizations of passive, common-ground, transformerless, two-elementkind RLC networks with minimum number of both nodes and elements of one kind can be generated with possibility of reduction in number of elements of other kind.The network synthesis is always performed by using normalization of both the complex frequency s and the impedance Z s .If Ω is a selected normalization frequency, then the normalized frequency is s n s/Ω.Similarly, if R 0 is a selected normalization resistance, then the normalized impedance is Z n s Z s /R 0 .Thereby we achieve 20 : a lesser dispersion of coefficients in normalized functions and b dimensionless manipulation of quantities.The normalized resistance of resistor R is R n : R/R 0 .The normalized impedance of an inductor L is To physically realize a network after synthesis, a denormalization process must be performed.The actual parameter values of RLC elements are calculated as follows: R . From now on it will be assumed that normalized synthesis is being carried out, but the lower index "n" we be dropped from component labels for brevity.
It is well known that if a real rational function in s can be realized as RL drivingpoint impedance Z RL s , then it can be also realized as RC driving-point admittance Y RC s 20 .And similarly, if it can be realized as RL driving-point admittance Y RL s , then it can also be realized as RC driving-point impedance Z RC s .The LC : RC transformation turns the synthesis of LC driving-point impedance Z LC s to synthesis of RC driving-point impedance These RC driving-point imittances are at first realized by prototype RC networks and thereof are produced the desired LC networks in the following way: capacitors in RC and LC networks remain the same, but the resistor from RC network turns to inductor in LC network with the same parameter value.Also, LR : RC transformation turns the synthesis of RL drivingpoint impedance Z RL s to synthesis of RC driving-point impedance Z RC s Z RL s /s.It also turns synthesis of RL driving-point admittance Y RL s to synthesis of RC driving-point admittance Y RC s sY RL s .These RC imittances are realized by prototype RC networks and the desired RL networks are produced thereof in the following way: the resistor from RC network turns to inductor in LR network with the same parameter value, and the capacitor from RC network turns to resistor in LR network with reciprocal parameter value.Bearing all the aforementioned on mind, we can obviously restrict our consideration only to synthesis of driving-point impedance functions Z RC s of RC networks, which satisfy the following well known analytic necessary and sufficient conditions 20 : a Z RC s is real rational function in s, b It has only simple poles on negative real axis, or at the origin.At infinity it cannot have pole and c Residues of these poles are real and positive and A ∞ : lim s→∞ Z RC s ≥ 0.
In general, the first canonic Foster's expansion form of Z RC s 20 reads where A m is residue of the pole s m m 0, 1, . . ., n .The network which realizes drivingpoint impedance Z RC s 5.1 with minimum number of nodes n 1 , minimum number of resistors n 1 and minimum number of capacitors n 1 is depicted in Figure 1.Observe that neither the resistors, nor the capacitors share common-node and hence the overall network realization is said to be non common-grounded.Now, we will present our synthesis procedure.If for a given driving-point impedance Z RC s we found that A 0 > 0 and/or A ∞ > 0, then in the preamble of the realization Mathematical Problems in Engineering The first canonic Foster's realization of Z RC s .Denoted are the normalized "values" of RC parameters.
procedure A ∞ and/or A 0 /s should be at first extracted from 5.1 and realized by a series connection of resistor R ∞ A ∞ and capacitor C 0 1/A 0 , thereby leaving for realization the driving-point impedance Z RC s Z RC s − A ∞ − A 0 /s with solely n poles lying on the negative real axis.In the sequel we will assume that Z RC s has only n such poles. Let Various network topologies can be produced by different choices of U and V. But, only by selecting C CI n C > 0 and V I n , the networks with minimum number of commonground capacitors are produced; and only by selecting G GI n G > 0 and V I n , the networks with minimum number of common-ground resitors are produced.Let us select C CI n C > 0 and V I n , and let us assume in 5.1 : A 0 A ∞ 0 and s n > s n−1 , i 2, . . ., n .

5.4
To prove the existence of a physical realization of both Cδ 2 s and δ UGU T δ we still have to determine the positive column vector col δ δ 1 δ 2 then the column vector p of row column hd margins of matrix δ UGU T δ with hd sign pattern reads

5.6
This means that δ UGU T δ is truly hd.We will now present two algorithms for realization of driving-point impedances Z RC s which rely on the results developed above.
Algorithm 1. Realization of Z RC s with minimum number of common-ground capacitors and non-reduced number of resistors 1 0 Commencing with A i i 1, . . ., n calculate the entries of U, by using 2.26 and 5.4 .

A numerical example
Consider realization of the real rational function The reactance function corresponding to Z LC s is X LC ω : Z LC jω /j and it is depicted in Figure 2. The first canonic Foster's realization of Z LC s with minimum number of nodes, noncommon-ground capacitors and inductors is depicted in Figure 3. Thereon are denoted the normalized values of LC parameters.

Conclusions
In Figure 4, it is depicted the first canonic Foster's realization of driving-point impedance Z LC s from Figure 3 with selected normalization frequency Ω 10 6 rad/s and selected normalization resistance R 0 10 3 kΩ .The network is excited by a sinusoidal current generator having constant current amplitude and discretely varying frequency f ω/ 2π within the range f ∈ 0.1, 500 kHz.If the complex representative of generator current is I g and the complex representative of the voltage across its terminals is U g , then the complex driving-point impedance of the overall LC network is Z LC U g /I g .The modulus of Z LC , that is, |Z LC | usually called LC impedance obtained through PSPICE simulation within the range f ∈ 0.1, 500 kHz is depicted in Figure 5. Now, we will realize Z LC s by using the proposed Algorithm

6.1
In step 1 0 of Algorithm 2 we determine the orthogonal matrix U by using A 1 , A 2 , and A 3 see 2. By embedding in the third port a series connection of resistor and capacitor with the normalized parameter values 1 and 2/5, respectively, and by applying RC : LC transformation thereafter, we finally produce noncanonic network which realizes Z LC s with minimum number of nodes and capacitors and with reduced number of inductors.That network is depicted in Figure 6 whereon are denoted the normalized dimensionless values of LC parameters.
In Figure 7, is depicted the noncanonic realization of driving-point impedance Z LC s from Figure 6 with selected normalization frequency Ω 10 6 rad/s and selected normalization resistance R 0 10 3 kΩ .The network is excited by a sinusoidal current generator with constant current amplitude and with discretely variable frequency f ω/ 2π within the range f ∈ 0.1, 500 kHz.If the complex representative of generator current is I g and the complex representative of the voltage across its terminals is U g , then the complex driving-point impedance of the overall LC network is Z LC U g /I g .The modulus of Z LC , that is, |Z LC | usually called LC impedance obtained through PSPICE simulation within the range f ∈ 0.1, 500 kHz is depicted in Figure 8.Since LC networks in Figures 4 and  7 are intentionally designed to be equivalent, then their driving-point impedances |Z LC | must have the same variations in frequency, as can be verified from Figures 5 and 8 qualitatively and more precisely by using numerical results of simulation.
A novel procedure for explicit construction of entries of real symmetric matrices with assigned spectrum is developed by using a group of four types of canonic, secondorder, orthogonal transformations.It has been also shown that the orthogonal modal matrices corresponding to the produced real symmetrix matrices, are either lower or upper Hessenberg with explicitly constructed entries too.Thereafter, the inverse eigenvalue problems of real symmetric matrices with twelve specific types of sign patterns including hyperdominant one are explicitly solved providing that the signs of eigenvalues are the same zeros are permitted and that they are enumerated such as to establish the increasing or decreasing sequence.It is proved to arise thereof a possibility of explicit solving the inverse eigenvalue problem of symmetric hyperdominant matrices having either uncommitted or assigned nonnegative spectrum.The results obtained are then applied in synthesis of drivingpoint immittance functions of transformerless, common-ground, two-element-kind RLC networks and in generation of their equivalent realizations with minimum number of nodes.The synthesis procedures proposed herein turn the synthesis problem of any immittance function of the two-element-kind RLC network to the synthesis problem of impedance function of a prototype RC network.

Theorem 3 . 5 .
-2.18 , 2.21 , G −1 n can be calculated explicitly, also without really inverting G n .Let the positive increasing sequence {λ m } m 1, . . ., n be the spectrum of G n produced by using Case 1 of Theorem 3.3.Then there always exists such a diagonal matrix D : diag d 1 , d 2 , . . ., d n with positive diagonal entries which makes DG n D truly hyperdominant.Proof.If G 1 : diag λ 1 , λ 2 , . . ., λ n , then by Case 1 of Theorem 3.3, the nonsingular matrix G n UG 1 U T will have hd sign pattern and by Remark 3.4G −1 n UG −1

3 0
Calculate col δ μ u 11 u 21 • • • u n1T , by using 5.5 and the entries of hd matrix δ UGU T δ.Calculate p, by using 5.6 .40Realize δ UGU T δ by common-ground, transformerless, conductance network.This can be done easily, almost by visual inspection of δ UGU T δ 22 .Attach to the ports 4

Figure 2 :
Figure 2: The reactance function of LC network from the example.

δ 2 δ 3 T 2 .
step 2 0 , we further easily obtain C 1, G 1 2, G 2 4, G 3 6 and col δ δ 1 087 0.365 0.632 T .In step 3 0 we firstly calculate δ UGU T δ and then hd margins of its rows columns , calculate the normalized capacitances of common-ground capacitors: C 1 4.355, C 2 0.133 and C 3 0.400.Realization of driving-point impedance Z RC s by transformerless, common-ground RC network with minimum number of nodes n 1 , reduced number of inductors and minimum number of common-ground capacitors n , begins by realization of conductance matrix δ UGU T δ which can be accomplished almost by inspection of that matrix 22 .Then, to the ith port of the realized conductance network, it should be connected to the capacitor C i δ 2 i i 1, 2, 3 .The third port of the overall network 24 Mathematical Problems in Engineering realizes the RC driving-point impedance Z RC s , provided that all other ports are left opencircuited.
n − 1 .They are calculated according to the following steps: n − 1 are arbitrarily selected angles from the range 0, π/2 , then the entries of real symmetric matrices G n with assigned spectrum {λ 1 ,λ 2 , . . .,λ n }, produced by 2.21 , can attain the following twelve sign patterns (zero entries are permitted), depending on selection of matrices P k k 1, . . ., n − 1 (see 2.1 ).
is nonnegative and increasing but not strictly , matrix G n is produced with hd sign pattern, including the possible presence of zero entries.G n may attain a sparse structure if, for example, some eigenvalues are equal.To see that, let us firstly suppose λ 1• • • λ k λ.Then from 2.17 -2.18 it follows that λ * then the signs of a k and c k depend solely on selection of canonic orthogonal matrices P k k 1, . . ., n − 1 .For any sign of sequence {λ m } m 1, . . ., n and its monotonicy realized through enumeration of its members, one can readily check the sign patterns stated above: by using 2.18 to determine signs of the diagonal entries in G n and by using Lemma 3.1 or Lemma 3.2 to determine signs of ε k k 1, . . ., n − 1 .Observe that only in Case 1 when λ n ≥ λ n−1 ≥ • • • ≥ λ 1 ≥ 0, that is, when the sequence {λ m } m 1, . . ., n λ 2 , . . ., λ n with spectrum {λ 1 , λ 2 , . . ., λ n } and the entries determined by 2.21 , is produced with hd sign pattern, no matter what selection of θ k s k 1, . . ., n − 1 has been made.Observe that in Case 1 a k cos θ k and c k sin θ k .To make G n G n be diagonal n × n matrices with strictly positive diagonal entries corresponding to the normalized capacitances and conductances, respectively.If we arbitrarily choose a nonsingular n × n matrix T, then the reciprocal passive networks which realize Cs G and Y s T Cs G T T will have the same natural frequencies.By arbitrary selection of n × n nonsingular diagonal matrices δ diag δ 1 , δ 2 , . . ., δ n , a broad class of nonsingular n × n matrices T can be generated with assumption T VδU, where U and V are n × n orthogonal matrices.Since Y s • • • > s 1 > 0.The matrices which are effectively realized by common-ground network with n 1 nodes n 1 th node is the common-ground are Cδ 2 s and δ UGU T δ, provided that both are truly hd.According to 5.1 and 5.3 it holds G p Cs p p 1, ..., n andG n > G n−1 • • • > G 1 > 0. By using Theorem 3.3 Case 1, P k A k and θ k ∈ 0, π/2 k 1, . ..,n − 1 we infer that UGU T 2.21 is produced with hd sign pattern and no zero entries and with strictly positive inverse.Matrix U 2.26 is lower Hessenberg with nonnegative entries, except for negative "b"s.The same conclusions relating to UGU T and U also hold if we apply Case 1 of Theorem 3.3 with P k B k and θ k ∈ 0, π/2 k 1, . . ., n − 1 , except for "d"s in 2.26 are then negative and "b"s are positive.To realize Z RC s we must select in 5.P k A k and θ k ∈ 0, π/2 k 1, . . ., n − 1 , it follows from 2.26 , 5.1 , 5.3 • • • δ n T which, according to Theorem 3.5, makes δ UGU T δ truly hd with possibly zero hd margins of at most n − 1 rows.Let hd margin of the ith row in δ UGU T δ be p i i 1, ..., n and let p :p 1 p 2 • • • p nT .> 0 of arbitrarily assumed real nonegative numbers.Since UGU T −1 is strictly positive, then it always can be find a diagonal matrix δ diag δ 1 , δ 2 , . . ., δ n with positive diagonal entries, such that p * . . ., n , bearing on mind that at most n − 1 "p"s can be equal to zero.These "p"s indices correspond to indices of those rows or columns in δ UGU T δ which have zero hd margins.Then, from the overall network vanish resistors connecting common-ground to nodes with the same indices as that of rows columns with zero hd margins 22 .For different selections of p * , different algorithms and different topologically and parametrically equivalent realizations emerge.For example, if we select col δ μ u 11 u 21 • • • u n1 T μ > 0 , where it is according to 2.26 , 5.4 T T e 1 .Set for other hd margins p i 0 i 2, . . ., n .4 0 Realize δ UGU T δ by common-ground, transformerless, conductance network and attach to its ith port the common-ground capacitor with normalized capacitance Cδ 2 i i 1, . . ., n .The nth port of the overall network realizes driving-point impedance Z RC s , provided that all other ports are left open-circuited.
Z LC s as driving-point impedance of common-ground transformerless LC network with minimum number of capacitors and the reduced number of inductors This function satisfies the necessary and sufficient conditions for driving-point immittances of LC networks: a it is an odd real rational function in s; b it has only simple poles located strictly on imaginary axis; and c residues of those poles are real and positive.Therefore, Z LC s can be realized both in two Foster's and in two Cauer's canonic forms 20 .The partial fraction expansion of Z LC s reads 2. After LC : RC transformation, we firstly produce the function Z RC s Z LC √ s / √ s 1 5/2s Z RC s , where Z RC s is driving-point impedance of RC network which should be expanded into partial fractions as follows: , s 2 4, s 3 6, s 3 > s 2 > s 1 > 0, S : diag s 1 ,s 2 ,s 3 .