In this work is given a new pseudoniverse matrix method for balancing chemical equations. Here offered method is founded on virtue of the solution of a Diophantine matrix equation by using of a MoorePenrose pseudoinverse matrix. The method has been tested on several typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method, which works successfully without any limitations, also has the capability to determine the feasibility of a new chemical reaction, and if it is feasible, then it will balance the equation. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are introduced necessary and sufficient criteria for stability of chemical equations over stability of their extended matrices.
1. INTRODUCTION
Chemical equations are the most exploited part of theoretical as well as industrial chemistry. Every chemistry instructor doubtless has his favorite technique of balancing equations. Chemical equations are commonly written for two purposes: to show the probable reaction products, and to show the quantities of materials that enter into reaction. It is fortunate indeed that an extensive and comprehensive knowledge of subatomic phenomena is not essential in the balancing of equations. Every student, which has general chemistry as an obligatory subject, is bound to come across balancing chemical equations. It is highly important for chemistry student to be able to write and balance chemical equations readily and accurately. Actually, balancing chemical equations provides an excellent demonstrative and pedagogical example of interconnection between stoichiometrical principles and linear algebra.
A
chemical equation
is a symbolic representation of a
chemical reaction
. The substances initially involved in a
chemical reaction
are called
reactants
, but the newly formed substances are called the
products
. The products are new substances with properties that are different from those of reactants. Classically, chemical reactions encompass changes that strictly involve the motion of
electrons
in the forming and breaking of
chemical bonds
, although the general concept of a chemical reaction, in particular the notion of a
chemical equation
, is applicable to
transformations of elementary particles
.
A chemical equation is not only the shorthand writing of the chemist, but it should be a mental picture of an actual reaction. To the observant investigator, the equation should immediately remind him as to the physical nature and properties of the reactants,
viz
., color, state, etc., as well as the chemical result and its physical nature. Thus, a great deal of significance should be attached to the writing of chemical equations. The part of chemical mathematics called
Stoichiometry
deals with the weight relations determined by chemical equations and formulas. According to it, the balancing of chemical equations is important in this area. Since a chemical reaction, when it is feasible, is a natural process, the consequent equation is always consistent. Therefore, we must have a nontrivial solution and we should be able to obtain it assuming its existence. Such an assumption is absolutely valid and does not introduce any error. If the reaction is infeasible, then the only solution is a trivial one,
i. e.
, the all coefficients are equal to zero.
2. HISTORICAL SURVEY
During the past years, numerous articles have appeared in scientific literature relative to the balancing of chemical equations. In this section is made a short survey of selected articles for balancing chemical equations. The selection criteria for references were intentionally wide, in order to include a large variety of topics. Balancing chemical equations in the scientific literature is considered from four points of view: mathematical, computational, chemical and pedagogical.
Now, shortly we will describe these glances.
· Jones for the first time in mathematics proposed the general problem for balancing chemical equations.
1
Krishnamurthy in his article
2
gave an algebraic approach for balancing chemical equations founded on virtue of a generalized matrix inverse. Little bit late Das
3
offered a simple scholarly technique, which was discussed in.
4
,
5
One other technique for balancing chemical equation over an integer programming approach is given in,
6
while in
7
by using of a reflexive generalized inverse matrix is solved the general problem of balancing chemical equations given in.
1
The newest mathematical results for balancing chemical equations and their stability over a nonsingular matrix method are obtained in.
8
Actually, to date in mathematics and chemistry there are only three strictly formalized consistent mathematical methods for balancing chemical equations, particularly they are the methods given in
7
,
8
and right now presented method in this work, while other so called
methods
in chemical sense have a limited usage, and they are useful only for particular cases, especially for balancing chemical equations which possess atoms with integer oxidation numbers.
· In chemistry there are many published articles,
9

29
which consider the use of computers for balancing chemical equations. All of these computational methods use some commercial softer packet.
· University textbooks of general chemistry generally support the ionelectron technique as basic procedure for balancing chemical equation, because it makes the best use of fundamental chemical principles. Also, some authors advocated other techniques, which involve less algebraic manipulation that may deserve attention  particularly in classes of chemistry and chemical engineering majors.
30

56
Several simple chemical equations are solved by elementary algebraic techniques in.
45
,
57

62
Bottomley published the earliest article that makes use of the linear algebra method.
63
A set of various modifications, which implement this approach, is documented in.
31
,
44
,
45
,
64

66
The case when the chemical equation has no unique solution received considerable attention in the education articles.
18
,
67

79
The equation represents two or more independently occurring reactions can be combined in varying stoichiometric ratios.
80
,
81
Fixed ratios of reagents, observed experimentally in particular cases, are equivalent to a restriction on the coefficients that make a unique solution.
18
It is necessary to stress out that balancing chemical equations by inspection is equivalent to using the algebraic method or a computerized matrix algebra approach.
82
,
83
The valence change method
30
,
84

106
and the ionelectron method
85

88
,
92
,
100
,
107

113
are also simple algebraic inspection techniques, subjected to exactly the same controls and limitations as the algebraic and matrix methods. Here it is good to emphasis that first Karslake in
114
considered balancing of ionic chemical equations. Actually, the technique suggested by Garcia
115
can reduces the number of algebraic steps for ionelectron method. Above both mentioned methods  the valence change method and ionelectron method begin by establishing the relative proportions of reagents taking part in separate oxidation and reduction components of a redox reaction. Then, each technique uses a lowest common multiplier to enforce a principle of conservancy  for instance, conservation of oxidation number change in the case of the oxidation number method. Johnson in his article
116
defined the equivalent term
oxidation stage change
on this subject.
Stout in
117
presented three redox reactions as puzzles. Each one can be shown as simple redox system, which may easily be balanced using here offered method. After this article was published, the followed other debatable articles with critical accent.
118

123
· Balancing chemical equations through the pedagogical point of view is given in the articles.
112
,
124

133
This approach is very interesting for the education of chemical research. A check of the hypothesis that formal reasoning and a sufficiently large mental capacity are required to balance more complex manystep equations is made over a test to determine level of intellectual development, mental capacity, and degree of field dependence/field independence of the students.
134
3. PRELIMINARIES
Now we will introduce some well known results from the matrix algebra. Throughout, the set of
m
×
n
matrices over a field will be denoted by ℝ
^{m×n}
.
Definition 3. 1.
The MoorePenrose pseudoinverse A
^{+}
of a matrix
A
∈ℝ
^{m×n}
is the unique matrix satisfying the following criteria:
AA
^{+}
A
=
A
,
A
^{+}
AA
^{+}
=
A
^{+}
, (
AA
^{+}
)
^{T}
=
AA
^{+}
, (
A
^{+}
A
)
^{T}
=
A
^{+}
A
.
Suppose the matrix
A
has full rank, then
A
= [
a_{ij}
]
_{n×n}
and rank
A
=
n
⇒
A
^{+}
=
A
^{1}
,
A
= [
a_{ij}
]
_{m×n}
(
m
＞
n
) and rank
A
=
n
⇒
A
^{+}
= (
A
^{T}
A
)
^{1}
A
^{T}
,
A
= [
a_{ij}
]
_{m×n}
(
m
＜
n
) and rank
A
=
m
⇒
A
^{+}
=
A
^{T}
(
AA
^{T}
)
^{1}
.
If the matrix
A
does not have full rank,
i.e.
,
A
is an
m
×
n
matrix and rank
A
=
r
＜min(
m
,
n
), then pseudoinverse
A
^{+}
can be constructed from the singular value decomposition
A
=
UDV
^{T}
by
A
^{+}
=
VD
^{+}
U
^{T}
.
Also the MoorePenrose pseudoinverse
A
^{+}
of a matrix
A
, we can derive on this way. There do always exist two matrices
C
= [
c_{ij}
]
_{m×r}
and
D
= [
d_{ij}
]
_{r×n}
of rank
r
, such that
A
=
CD
. Using these matrices it holds that
A
^{+}
=
D
^{T}
(
DD
^{T}
)
^{1}
(
C
^{T}
C
)
^{1}
C
^{T}
.
Pseudoinverse matrix
A
^{+}
of a matrix
A
was independently defined by Moore
135
and Penrose
136
. If the matrix
A
^{+}
satisfies first two equalities of definition 3. 1, then it is called a
reflexive generalized inverse
of
A
, denoted by
A
^{–}
. This matrix is not unique.
Definition 3. 2.
The characteristic equation of an n
×
n matrix A is the equation in one variable
λ
where
det(·)
denotes a determinant and I is an n
×
n identity matrix
.
Definition 3. 3.
The polynomial
which results from evaluating the determinant
(3.1)
is the characteristic polynomial of the matrix A
.
The above polynomial of degree
n
≥1 with real coefficients
a
_{i}
(0 ≤
i
≤
n
), where
a_{n}
≠0, by the fundamental theorem of algebra has
n
(not necessarily distinct) roots
λ
_{1}
,
λ
_{2}
, … ,
λ
_{n}
.
Definition 3. 4.
The roots of the characteristic polynomial
(3. 2)
are precisely the eigenvalues of the matrix A
.
Definition 3. 5.
The reaction matrix has the following format A
¦
B
= [
A
¦
B
]
_{m×(r + s)}
,
where A
= [
a_{ij}
]
_{m×r}
(
r
＜
m
)
and B
= [
b_{ij}
]
_{m×s}
(
s
＜
m
).
Definition 3. 6.
An extended matrix
of
(
r
+
s
)×(
r
+
s
)
dimension, is singular if
det(Ext
A
¦
B
) = 0,
where A
= [
a_{ij}
]
_{m×r}
(
r
＜
m
),
B
= [
b_{ij}
]
_{m×s}
(
s
＜
m
),
I
is the identity matrix of
(
r
+
s

m
)×(
r
+
s

m
)
dimension and O is a zero matrix of
(
r
+
s

m
)×
m
dimension.
In
8
is treated the nonsingular case.
Let σ(Ext
A
¦
B
) = {
λ
_{i}
, 1≤
i
≤
k
} be the spectrum of Ext
A
¦
B
. Let  ·  denotes a vector norm in ℝ
^{k}
.
Definition 3. 7.
The LozinskiĬ measure
μ
on
ℝ
^{n} with respect to
 · 
is defined by
Definition 3. 8.
The LozinskiĬ measures of
Ext
A
¦
B
= [a
_{ij}
]
_{p×p}
with respect to the three common norms
are
where
stab(Ext
A
¦
B
) = max{
λ
,
λ
∈σ(Ext
A
¦
B
)}
is the stability modulus of
Ext
A
¦
B
and
T
denoting transpose
.
Definition 3. 9.
The extended matrix
Ext
A
¦
B
is stable if
stab(Ext
A
¦
B
) ＜ 0.
4. MAIN RESULTS
In this section we will give a completely new method for balancing and stability of chemical equations. Given analysis is done for arbitrary chemical equation presented in its general form.
Proposition 4. 1.
Any chemical equation may be presented in this form
where
x_{j}
(1≤
j
≤
r
)
and
y_{j}
(1≤
j
≤
s
)
are unknown rational coefficients, Ψ^{i} and Ω^{i}
(1≤
i
≤
m
)
are chemical elements in reactants and products, respectively, a_{ij}
(1≤
i
≤
m
; 1≤
j
≤
r
)
and b_{ij}
(1≤
i
≤
m
; 1≤
j
≤
s
)
are numbers of atoms of elements Ψ^{i} and Ω^{i}, respectively, in jth molecule
.
Proof
. Let there exists an arbitrary chemical equation from
s
distinct elements and
m
+
n
molecules
where Ξ
_{j}
=Ψ
^{1}
_{a1j}
Ψ
^{2}
_{a2j}
···Ψ
^{m}_{amj}
(1≤
j
≤
r
) and Θ
_{j}
= Ω
^{1}
_{b1j}
Ω
^{2}
_{b2j}
···Ω
^{m}_{bmj}
(1≤
j
≤
s
). Then previous expression becomes
Now, if we write the above equation in a compact form, then immediately follows (4.1).
The coefficients
x_{j}
,
y_{j}
satisfy three basic principles (corresponding to a closed inputoutput static model
137
,
138
)
· the low of conversation of atoms,
· the low of conversation of mass, and
· the timeindependence of the reaction.
Theorem 4. 2.
The chemical equation
(4.1)
can be reduced to the following Diophantine matrix equation
where A
=[
a_{ij}
]
_{m×r}
is a matrix of the reactants, B
=[
b_{ij}
]
_{m×s}
is a matrix of the products
,
x
^{T}
=(
x
_{1}
,
x
_{2}
, …,
x_{r}
)
and y
^{T}
=(
y
_{1}
,
y
_{2}
,…,
y_{s}
)
are column vectors of unknown coefficients and T denotes transpose
.
Proof
. If we develop the molecules of the reaction (4.1) in an explicit form, then we obtain a matrix of the reactants
A
and a matrix of the products
B
From the above development we obtain that
and
If we substitute (4.5) and (4.6) into (4.2), follows
or
i.e.
, the above expression holds if only if
Last equation if we present in a matrix form, actually we obtain (4. 4).
Now we will prove the following result.
Theorem 4. 3.
Let A
∈ℝ
^{m×r}
and
B
∈ℝ
^{m×s}
.
The matrix Diophantine equation
(4. 4)
is consistent if and only if for some A
^{+}
where
If this is the case, a representation of the general solution is
and
with arbitrary vectors u
∈ ℝ
^{s×1}
and v
∈ ℝ
^{r×1}
.
Proof
. Matrix equation (4. 4) is consistent if and only if there exist vectors
x
∈ℝ
^{r×1}
and
y
∈ℝ
^{s×1}
such that
From (4. 13) we have
Ax
=
AA
^{+}
By, since
A
(
I

A
^{+}
A
) = 0. Thus, from (4. 14) follows
On the other hand, from (4. 12) it follows that
Immediately, from (4.16) and (4.15) follows (4.4).
Remark 4. 4.
This theorem generalizes the theorem proved in7. By the above theorem, a century old chemistry problem of balancing chemical equations in a general form is completely solved
.
Definition 4. 5.
Chemical equation
(4.1)
is stable if
stab(Ext
A
¦
B
) ＜ 0.
Lemma 4. 6.
For any nonsingular matrix U and any vector norm
│ · │,
with the induced Lozinski measure
μ, │
Ux
│
defines another vector norm and its induced matrix measure
μ_{U} is given by
Proof
. The proof of this lemma follows directly from the definition 3.7.
Theorem 4. 7.
For any matrix
Ext
A
¦
B
∈ℝ
it holds
Proof
. The relation (4.18) obviously holds for diagonizable matrices in view of (4.17) and the first two relations in (3.6). Furthermore, the infimum in (4.18) can be achieved if the matrix Ext
A
¦
B
is diagonizable. The general case can be shown based on this observation, the fact that Ext
A
¦
B
can be approximated by diagonizable matrices in and the continuity of μ(·), which is implied by the property
μ(
ℑ
)  μ(
ℵ
) ≤ 
ℑ

ℵ
.
Remark 4. 8.
From the above proof it follows that
stab(Ext
A
¦
B
)=inf{μ
_{∞}
[
U
(Ext
A
¦
B
)
U
^{1}
], det
U
≠0}.
The same relation holds if μ is replaced by μ
_{1}
.
Corollary 4. 9.
Let
Ext
A
¦
B
∈ℝ.
Then
stab(Ext
A
¦
B
)＜0⇔μ(Ext
A
¦
B
) ＜ 0
for some LozinskiĬ measure
μ
on
ℝ
^{n}
.
There are more results for stability criteria obtained in the works.
139
,
140
For the matrices
A
and
B
given in the theorem 4.2, let
rankA
¦
B
=
ρ
and
ker
A
¦
B
=
κ
, where
ker
denotes the
nullity
or kernel of matrix
A
¦
B
. According to
141
, the deterministic approach is important, since it enables us to classify the chemical reaction as:
1˚
infeasible
when the nullity of the reaction matrix
A
¦
B
is zero;
2˚
unique
(within relative proportions) when the nullity of the reaction matrix
A
¦
B
is one; or
3˚
nonunique
when the nullity of the reaction matrix
A
¦
B
is bigger than one.
Possible cases of balancing chemical equations are the following
1. If
m
≥
r
+
s
=
ρ
, then
κ
=
r
+
s

ρ
= 0,
i. e
., trivial solution
x, y
=
0
, the reaction is infeasible.
2. If
m
≥
r
+
s
,
ρ
=
r
+
s
 1, then
κ
=
r
+
s

ρ
= 1, unique solution
x, y
＞
0
,
i. e.
, the reaction is feasible and is unique.
In practical terms this means that the general procedure for obtaining these coefficients is to solve the system of linear equations derived from the principles of conservation of matter and charge, applied to the reaction elementbyelement.
3. If
m
≥
r
+
s
,
ρ
＜
r
+
s
 1, then
κ
=
r
+
s

ρ
＞ 1,
κ
(＞1) linearly independent solutions
x, y
＞
0
,
i. e
., the reaction is feasible and is nonunique.
4. If
m
＜
r
+
s
,
ρ
≤
m
, then
κ
=
r
+
s

ρ
≥1,
κ
(≥1) linearly independent solutions
x, y
＞ 0,
i. e.
, the reaction is feasible and is nonunique.
Last two kinds of the reactions are puzzling in that they exhibit infinite linearly independent solution all of which satisfy the chemical balance, and yet they are not all chemically feasible solutions for a given set of experimental conditions. A unique solution is obtained by imposing a chemical constraint, namely, that reactants have to react only in certain proportions.
5. AN APPLICATION OF THE MAIN RESULTS
In this section will be applied above method on many chemical equations for their balancing. All chemical equations balanced here appear first time in professional literature and they are chosen with an intention to be avoided to date all well know chemical equations which were repeated many times in the chemical journals for explanation of certain particular techniques for balancing of some chemical equations using only atoms with integer oxidation numbers.
1˚ First we will consider an infeasible reaction,
i. e.
, the case when the nullity of the reaction matrix is zero.
Example 5. 1.
Consider this equation
By the schemes
are determined the matrix of reactants
A
and the matrix of the products
B
, an according to it the reaction matrix
A
¦
B
obtains the form
The rank of the above matrix is
ρ
= 11. Since the nullity of the reaction matrix is
κ
=
r
+
s

ρ
= 5 + 6  11 = 0, then we have only a trivial solution
x, y
=
0
, that means that the reaction is infeasible.
2˚ Next, we will consider the case when the chemical reaction is feasible and is unique,
i. e.
, the nullity of its reaction matrix is one. Here we will balance many special chemical equations with a goal to show the power of the offered mathematical method.
Example 5. 2.
As a next example to illustrate above method, we will balance this chemical equation
From the schemes given below
are determined the matrix of reactants
A
and the matrix of the products
B
,
i. e.
,
The reaction matrix
A
¦
B
, according to the definition 3. 5 obtains the form
The rank of the above matrix is
ρ
= 9. Since the nullity of the reaction matrix is
κ
=
r
+
s

ρ
= 3 + 7  9 = 1, then we have only a unique solution, that means that the reaction is feasible. Next, we will determine the solution
x, y
.
The MoorePenrose pseudoinverse
A
^{+}
= (
A
^{T}
A
)
^{1}
A
^{T}
of the matrix
A
is
and the matrix
G
has this form
688
G
= (688
I

AA
^{+}
)
B
=
Required vectors
y
and
x
, according to (4.12) and (4.13) are
Now immediately follows balanced equation in its conventional form
0.1356805781 Pt(NH
_{3}
)BrNO
_{3}
+0.5130378112 Cu(NH
_{3}
)KNO
_{3}
+1.5338181820 BeCO
_{3}
= 0.0348595041 C
_{44}
H
_{50}
O
_{14.98}
+0.5130378112 Cu(NO
_{3}
)
_{2}
+0.1356805781 PtO
_{3}
+0.0681811950 Br
_{1.99}
NO
_{2}
+0.2604252849 K
_{1.97}
O+1.5338181820 BeO+0.2031799611 HNO
_{3}
.
Since the reaction matrix
A
¦
B
has a rectangular format, according to definition 3. 6, we will extend it to a square matrix by adding a new row. To do that, we will add the following vectorrow (0, 0, 0, 0, 0, 0, 0, 0, 0, 1) as a 10th row, such that Ext
A
¦
B
obtains this form
The eigenvalues of the matrix [Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2 are
λ
_{1}
=35.111,
λ
_{2}
=33.595,
λ
_{3}
=7.769,
λ
_{4}
=1.630,
λ
_{5}
= 1.140,
λ
_{6}
=0.919,
λ
_{7}
=0.205,
λ
_{8}
=1.285,
λ
_{9}
=1.087,
λ
_{10}
= 0.807.
The LozinskiĬ measures of Ext
A
¦
B
given by (3.6) with respect to the three common norms (3.5) are
μ
_{∞}
(Ext
A
¦
B
) = max (2, 8, 57, 2.99, 36.98, 2, 2.97, 2, 45, 1) = 57,
μ
_{1}
(Ext
A
¦
B
)=max (10, 10, 5, 108.98, 9, 4, 4.99, 2.97, 2, 6)=108.98,
μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} = 35.111.
Since μ
_{2}
{[Ext
A
¦
B
+(Ext
A
¦
B
)
^{T}
]/2}>0, according to the definition 4.5 the chemical equation (5.2) is unstable.
Example 5. 3.
The chemical equation
like a characteristic case will be balanced too.
From the schemes given below
are determined the matrix of reactants
A
and the matrix of the products
B
.
The reaction matrix
A
¦
B
, according to the definition 3. 5 obtains this form
The rank of the above matrix is
ρ
= 10. Since the nullity of the reaction matrix is
κ
=
r
+
s

ρ
= 5 + 6  10 = 1, then we have only a unique solution, that means that the reaction is feasible. Next, we will determine the solution
x, y.
The MoorePenrose pseudoinverse
A
^{+}
=(
A
^{T}
A
)
^{1}
A
^{T}
of the matrix
A
is
3821829615
A
^{+}
=
and the matrix
G
has this form
3821829615
G
= (3821829615
I

AA
^{+}
)
B
=
Required vectors
y
and
x
, according to (4.12) and (4.13) are
Now immediately follows balanced equation in its conventional form
1.0653080872 Pt(NH
_{3}
)BrNO
_{3}
+0.0208499800 [4Fe(CN)
_{3}
·3Fe(CN)
_{2}
] + 0.0320088416 C
_{44}
H
_{50}
O
_{15}
+ 1.2365580746 Cu(NO
_{3}
)
_{2}
+ 0.9079895101 K
_{2}
BeO
_{2}
=1.2365580746 Cu(NH
_{3}
)KNO
_{3}
+0.9079895101 BeCO
_{3}
+ 0.1459498604 K
_{3.97}
Fe(CN)
_{6}
+ 1.0653080872 PtO
_{2}
+ 0.5435245343 Br
_{1.96}
NO
_{2}
+ 1.0866921189 HNO
_{3}
.
According to the definition 3. 6, extended matrix obtains the form
The eigenvalues of the matrix [Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2 are
λ
_{1}
=59.824,
λ
_{2}
=21.786,
λ
_{3}
=10.767,
λ
_{4}
=5.648,
λ
_{5}
=4.030,
λ
_{6}
=2.045,
λ
_{7}
=0.940,
λ
_{8}
=0.044,
λ
_{9}
=1.678,
λ
_{10}
=0.831,
λ
_{11}
=1.059.
The LozinskiĬ measures of Ext
A
¦
B
given by (3.6) with respect to the three common norms (3.5) are
μ
_{∞}
(Ext
A
¦
B
) = max (2, 32, 57, 2.96, 39, 8, 69, 2, 6.97, 2, 1) = 69,
μ
_{1}
(Ext
A
¦
B
) = max (10, 43, 109, 9, 5, 10, 5, 16.97, 3, 4.96, 6) = 109,
μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} = 59.824.
Since μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} > 0, according to the definition 4.5 the chemical equation (5.3) is unstable.
Example 5. 4.
We will now balance this equation
The matrix of reactants
A
is
and the matrix of the products
B
is
The reaction matrix
A
¦
B
, according to the definition 3. 5 obtains this form
The rank of the above matrix is
ρ
= 15. Since the nullity of the reaction matrix is
κ
=
r
+
s

ρ
= 8 + 8  15 = 1, then we have only a unique solution, that means that the reaction is feasible. Next, we will determine the solution
x, y.
The MoorePenrose pseudoinverse
A
^{+}
=(
A
^{T}
A
)
^{1}
A
^{T}
of the matrix
A
is
94083728
A
^{+}
=
and the matrix
G
has this form
94083728
G
= (94083728
I

AA
^{+}
)
B
=
Required vectors
y
and
x
, according to (4.12) and (4.13) are
Now immediately follows balanced equation in its conventional form
3.0772746861 KAu(CN)
_{2}
+0.0710721984 AgRuAuTe
_{8}
+ 0.3865922972 Fe
_{2}
(SO
_{4}
)
_{3}
+ 0.0339665957 N
_{2}
Se
_{4}
+ 0.1358663829 WO
_{3}
+ 0.1623687648 Na
_{2}
CO
_{3}
+ 1.1170850360 H
_{2}
CO
_{3}
+ 0.8214763117 HCl = 0.0710721984 [Ru(C
_{10}
H
_{8}
N
_{2}
)
_{3}
]Cl
_{2}
·6H
_{2}
O + 0.1656824131 C
_{4}
H
_{3}
AuNa
_{1.96}
OS
_{7}
+ 0.0679331915 [WCl
_{4}
(NSeCl)]
_{2}
+ 0.7731845945 K
_{3.98}
Fe(CN)
_{6}
+ 1.4913322357 Au
_{2}
O
_{3}
+ 0.5685775876 TeO
_{3}
+ 0.0710721984 AgO + 1.0210754230 NO
_{2}
.
According to the definition 3. 6, extended matrix obtains the form
The eigenvalues of the matrix [Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2 are
λ
_{1}
=28.656,
λ
_{2}
=24.826,
λ
_{3}
=9.278,
λ
_{4}
=6.650,
λ
_{5}
=6.847,
λ
_{6}
=5.301,
λ
_{7}
=4.806,
λ
_{8}
=3.495,
λ
_{9}
=3.389,
λ
_{10}
=2.805,
λ
_{11}
=1.297,
λ
_{12}
=0.937,
λ
_{13}
=0.759,
λ
_{14}
=0.312,
λ
_{15}
=0.040,
λ
_{16}
=0.134.
The LozinskiĬ measures of Ext
A
¦
B
given by (3.6) with respect to the three common norms (3.5) are
μ
_{∞}
(Ext
A
¦
B
) = max (4.98, 5, 44, 19, 2, 2, 9, 3, 10, 37, 6, 3, 3.96, 42, 13, 1) = 44,
μ
_{1}
(Ext
A
¦
B
) = max (6, 11, 17, 6, 4, 6, 6, 2, 81, 17.96, 16, 16.98, 5, 4, 2, 4) = 81,
μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} = 28.656.
Since μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} > 0, according to the definition 4. 5 the chemical equation (5.4) is unstable.
Example 5. 5.
Also, we balanced this very interesting chemical equation
According to the definition 3. 6, extended matrix obtains the form
The eigenvalues of the matrix [Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2 are
λ
_{1}
=143.364,
λ
_{2}
=41.450,
λ
_{3}
=14.630,
λ
_{4}
=11.150,
λ
_{5}
=11.154,
λ
_{6}
=6.730,
λ
_{7}
=4.060,
λ
_{8}
=3.424,
λ
_{9}
=3.357,
λ
_{10}
=3.432,
λ
_{11}
=2.112,
λ
_{12}
=2.481,
λ
_{13}
=1.348,
λ
_{14}
=1.453,
λ
_{15}
=0.544,
λ
_{16}
=0.614,
λ
_{17}
=0.575,
λ
_{18}
=0.137,
λ
_{19}
=0.090,
λ
_{20}
=0.006.
The LozinskiĬ measures of Ext
A
¦
B
given by (3.6) with respect to the three common norms (3.5) are
μ
_{∞}
(Ext
A
¦
B
) = max (377, 296, 4, 35, 85, 9, 6, 3, 2, 3, 9.986, 15, 2, 4, 2, 2, 13.97, 4, 4, 1) = 377,
μ
_{1}
(Ext
A
¦
B
) = max (168, 86, 95, 88, 90, 29, 2, 5, 3, 3, 85, 85, 98, 6, 11.986, 3, 3, 7.97, 3, 6) = 168,
μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} = 143.364.
Since μ
_{2}
{[Ext
A
¦
B
+(Ext
A
¦
B
)
^{T}
]/2}>0, according to the definition 4. 5 the chemical equation (5.5) is unstable.
Example 5. 6.
An other chemical equation which we balanced is the equation
The reaction matrix
A
¦
B
, according to the definition 3.5 has the form
The eigenvalues of the matrix [
A
¦
B
+(
A
¦
B
)
^{T}
]/2 are
λ
_{1}
=131.840,
λ
_{2}
=24.264,
λ
_{3}
=15.402,
λ
_{4}
=13.779,
λ
_{5}
=8.097,
λ
_{6}
=4.189,
λ
_{7}
=3.564,
λ
_{8}
=3.485,
λ
_{9}
=2.396,
λ
_{10}
=1.459,
λ
_{11}
=1.347,
λ
_{12}
=1.023,
λ
_{13}
=0.923,
λ
_{14}
=0.748,
λ
_{15}
=0.449,
λ
_{16}
=0.722,
λ
_{17}
=0.513,
λ
_{18}
=0.116,
λ
_{19}
=0.011,
λ
_{20}
=0.075.
The LozinskiĬ measures of
A
¦
B
with respect to the three common norms are
μ
_{∞}
(
A
¦
B
) = max (239, 195, 6, 8, 86, 12, 8, 3, 2, 2, 3, 2.99, 10, 4, 8, 3, 7.96, 3, 3, 3) = 239,
μ
_{1}
(
A
¦
B
) = max (158, 88, 13, 20, 7, 2, 5, 3, 4, 7, 3, 152, 83, 41.96, 5, 2, 3, 4.99, 4, 3) =158,
μ
_{2}
{[
A
¦
B
+ (
A
¦
B
)
^{T}
]/2} = 131.840.
Because μ
_{2}
{[
A
¦
B
+(
A
¦
B
)
^{T}
]/2}>0, that means that the chemical equation (5. 6) is unstable.
Example 5. 7.
It will be benefit for us if we balance some little bit bigger chemical equation, for instance like this
According to the definition 3.6, extended matrix obtains the form
The eigenvalues of the matrix [Ext
A
¦
B
+(Ext
A
¦
B
)
^{T}
]/2 are
λ
_{1}
=112.534,
λ
_{2}
=45.894,
λ
_{3}
=22.466,
λ
_{4}
=22.572,
λ
_{5}
=11.987,
λ
_{6}
=11.234,
λ
_{7}
=5.757,
λ
_{8}
=5.367,
λ
_{9}
=5.011,
λ
_{10}
=3.199,
λ
_{11}
=2.849,
λ
_{12}
=2.010,
λ
_{13}
=1.948,
λ
_{14}
=1.549,
λ
_{15}
=1.571,
λ
_{16}
=1.269, λ
_{17}
=1.042, λ
_{18}
=1.070, λ
_{19}
=0.814, λ
_{20}
=0.614, λ
_{21}
=0.809, λ
_{22}
=0.516, λ
_{23}
=0.398, λ
_{24}
=0.154, λ
_{25}
=0.091, λ
_{26}
=0.012.
The LozinskiĬ measures of Ext
A
¦
B
given by (3.6) with respect to the three common norms (3.5) are
μ
_{∞}
(Ext
A
¦
B
) = max (222, 210, 3, 26, 84, 3, 39, 5.97, 3, 11, 2, 13, 2, 6, 5, 3, 7, 2, 2, 4, 2, 2, 2, 4, 8, 1) = 222,
μ
_{1}
(Ext
A
¦
B
) = max (137, 16, 5, 6, 3, 68, 5, 4, 8, 17, 4, 5, 4, 13, 138, 1, 151, 12, 6, 4, 14, 8, 8, 23, 7.97, 4) = 151,
μ
_{2}
{[Ext
A
¦
B
+ (Ext
A
¦
B
)
^{T}
]/2} = 112.534.
Since μ
_{2}
{[Ext
A
¦
B
+(Ext
A
¦
B
)
^{T}
]/2}>0, according to the definition 4. 5 the chemical equation (5.7) is unstable.
Example 5. 8.
Also we balanced this chemical equation
Also, the above equation is unstable.
Example 5. 9.
This balanced chemical equation
is unstable too.
3˚ Next, we will consider the case when the chemical reaction is nonunique,
i. e.
, when the nullity of its reaction matrix is bigger than one.
Example 5. 10.
Last considered equation in this work will be this equation
From (5. 10) follows this system of linear equations

x1+x2=y2+ 2y3+ 2y4,

4x1+x2+x3=y1+ 2y5,

x1+x3=y1+y2+ 2y6,

4x1+ 3x2= 4y1+y2+y3+ 3y4+y5.
The general solution of the above system is

y1= 4x1+x2+x3 2y5,y2=  3x1x2+ 2y5 2y6,

y3= 15x1/2 + 3x2/2 + 2x3 4y5+y6/2,

y4=  11x1/2 x2/2  2x3+ 3y5+y6/2.
where
x
_{1}
,
x
_{2}
,
x
_{3}
,
y
_{5}
and
y
_{6}
are arbitrary real numbers.
Now, the balanced equation has a form

x1NH4ClO4+x2HNO3+x3HCl = (4x1+x2+x3 2y5) HClO4+ ( 3x1x2+ 2y5 2y6) NOCl + (15x1/2 + 3x2/2 + 2x3 4y5+y6/2) N2O + ( 11x1/2 x2/2  2x3+ 3y5+y6/2) N2O3+y5H2O +y6Cl2.
where
x
_{1}
,
x
_{2}
,
x
_{3}
,
y
_{5}
and
y
_{6}
are arbitrary real numbers.
A particular case of equation (5. 10) for
y
_{4}
= 0 is solved in8, but there the author balanced this equation on completely different way by using of nonsingular matrix method. So, it is the mathematical consideration of this equation, but for chemical purposes stoichiometric coefficients must be unique.
Now, if we employ here offered method we obtain that only possible chemical equation is this equation

30 NH4ClO4+ 2 NOCl + 12 Cl2= 11 HClO4+ 6 N2O + 5 N2O3+ 10 HNO3+ 45 HCl + 27 H2O,
which actually is a particular case of the chemical equation (5.10) for
x
_{1}
=1,
x
_{2}
=1/3,
x
_{3}
=3/2,
y
_{5}
=9/10 and
y
_{6}
=2/5.
Also, in the article
78
, is balanced a particular case for
x
_{1}
=34,
x
_{2}
=36,
x
_{3}
=8, and
y
_{2}
=
y
_{6}
=0.
6. CONCLUSION
The practical power of the matrix procedure as the most general tool for balancing chemical equations is demonstrable. By this method are balanced completely new classes of chemical equations with atoms, which have fractional oxidation numbers. Research shown that employed pseudoinverse matrix method founded on virtue of the solution of the Diophantine matrix equation works perfectly for the all chemical equations.
Actually, this method is unique method both in mathematics and chemistry, which generalize to date the all known methods and particular techniques for balancing chemical equations cited in the bibliography given below. In other words speaking, the mathematical method given here is applicable for the all possible cases for balancing chemical equations, does not matter what kind of atoms they possess – fractional or integer oxidation numbers. Also, this method determine and
r
+
s

ρ
linear independent solutions when
κ
> 1,
i.e.
, when the reaction is feasible and is nonunique. This is the main advantage of the method in relation of other known particular methods and techniques.
For all considered chemical equations which have a unique solution is made stability analysis, and as shown results all of them are unstable. This stability analysis is founded on virtue of the Lozinski measures of extended matrix.
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