_______________________________________________________________________________

In order to solve a set of linear equations, one proceeds by using a system called matrices. Matrices are arrays of numbers representing the coefficients of a linear system’s equations.

Here are some examples of matrices:

In the Holy Name of God, the All-Compassionate, the All-Merciful,

With peace and blessings upon His Prophet Muhammad SAW

upon his holy family, and upon his blessed Companions.

Bridging the Gap between Spirituality and Science

Matrices have different spiritual meanings that vary depending on the level of interpretation of the seeker. In linear algebra, they are the manifestation of the reality of sainthood.

The attentive reader may notice that earlier we described linear algebra, which is the field encompassing matrices, as a manifestation of the knowledge of God. However, we just established the fact that matrices are related to the knowledge of the saints or fiends of God, as they are referred to in the Holy Quran. How did we go from knowledge of God to knowledge of His friends? The answer is found in the following saying of the Prophet :

This saying implies that knowledge of the part (reality of the human self) will lead to knowledge of the whole (reality of God). Hence, by establishing a connection between matrices and the saints of God who are the ones who unlocked the knowledge of their true selves, one may better understand the science of linear algebra as it pertains to the knowledge of God.

When we are given a matrix, we try to solve it by reshaping it into a particular mathematical form called the echelon form. The matrix has to follow a set of mathematical rules in order for it to be in the echelon form. These mathematical rules are critical in solving the system of equations the same way the spiritual rules of a spiritual path are critical in reaching the level of knowledge of God. Here is an example of a matrix in the echelon form:

The idea behind the echelon form is to have 0’s under each leading entry of the matrix. Also, each leading entry must be in a row below another leading entry and to the right of that leading entry so that the matrix looks like a step-like pyramid, thus the name “echelon form”. The properties of a matrix in the echelon form are as follows:

- Property 1: all entries in a column below a leading entry are zeros

- Property 2: each leading entry of a row is in a column to the right of the leading entry of the row above it.

- Property 3: all nonzero rows are above any rows of all zeros (since such a row is null, it may be deleted from the matrix).

The most important entry (or number) in a matrix is the leading entry of a row. The leading entry in each nonzero row is called a “pivot”. In figure 12, the pivots are 3, 4, 7 and 3. Spiritually speaking, the pivots represent the saints of God. In mathematics, the pivots are the keys to solving the system of equations and to finding the unknown variables. The mathematical pivot is the manifestation of the saint who is the key to gaining knowledge of God.

Each pivot is in a different column called “pivot column”. These pivot columns represent the spiritual path taken by each saint (or pivot). Each path represents a set of increasing levels of spirituality. These levels are the rows of the matrix. The top row corresponds to the level of spirituality that is closest to heaven while the bottom row corresponds to the level of spirituality that is closest to creation.

The three properties of the echelon form mentioned earlier are the manifestations of some of the spiritual rules pertaining to the reality of sainthood. Let’s take interpret them in a spiritual context.

The key word in this property is “zeros”. The property means that in each column containing a pivot, there can only be null numbers below this pivot. These null numbers represent the spiritual followers of the saint.

At the top of each column (i.e., at the beginning of a each spiritual path) is the pivot (the saint). According to property 1 all entries below the pivot must be null because the spiritual seekers who are following a saint must show complete submission. This station is the station of “walking in the footsteps of the saint”. The seekers must have a zero spiritual level compared to the saint who is above their heads. They must show perfect humility and non-existence in order to gain access to the knowledge of God. In other words, in order for the system to have a solution, all entries below the pivots must be zeros.

Property 2 stipulates that each pivot must be located to the right of the pivot above it. This property is a manifestation of the holy verse:

Each saint (or pivot) stands below and at the right of the saint that is closer to Divine Presence, thus acknowledging the spiritual superiority of the saint above him.

Property 3 states that all nonzero rows are above any rows of all zeros. This property is a manifestation of the verse quoted above also. Each row in a matrix corresponds to a spiritual level. Therefore, the nonzero rows, which are high spiritual levels, are above the rows of all zeros, which are lower spiritual levels.

Once the matrix is in the echelon form it is easy to answer the first fundamental question of linear algebra. Mathematically speaking the system is said to have a solution if and only if there exists a pivot in the last row of the matrix (excluding the rightmost entry). This law is called the existence theorem of linear algebra. Here we assume that every other row has a pivot so that the last row is the only one that needs to be checked for the existence of a pivot.

Conversely, the system has no solution if the last row of the matrix is of the form:

0 0 0 0 0 6 or 0 0 0 0 0 7

The system has no solution because if we exclude the rightmost entries of the matrices (6 and 7) we see that there is no pivot since all the numbers are null. Note the existence theorem excludes the rightmost entries of the augmented matrix while checking for the existence of a pivot because these entries are not part of the coefficients of the system of equations that determine the unknown variables. So for the matrix to be solvable there must be a pivot in the last row.

The pivot in the last row of the matrix represents the saint who is among society and who connects creation with the higher saints who are in the Divine Presence.

As we see from the matrix, in spirituality, all those who follow the guide of a spiritual path (the columns of the matrix) exhibit signs of complete non-existence (the zeros in the columns). Moreover, the existence theorem guarantees the presence of a saint among a community who establishes a link between the community and the higher saints who are in the Divine Presence.

This living saint who mingles with common folks is the most important saint in the way of spirituality because of the pivotal nature of his station which connects human beings with the higher levels of spirituality. If he is not present on the spiritual path then the chain of transmission of divine knowledge is broken. Similarly, if the last pivot is not present, the system has no solutions.

In the realm of comparative religious studies, the last pivot could be identified as the first man, Adam, who was put on earth to represent God and gain knowledge about His Attributes. As the prototype of human beings, Adam is connected to creation through his own physical nature of water and clay and to Heaven through the prophetic or adamic light he was chosen to bear.

This pivotal nature is expressed in the following Verses:

and

and

No-solution systems are the manifestation of creation’s inability to know about the Essence of God. Since the solution of a mathematical equation is the manifestation of the knowledge of God through His Attributes, a no-solution equation means that God remains hidden and unknown.

An example of a no-solution system of equations would be:

One may see that there is no pivot in the last row of the matrix (excluding the rightmost entry since this entry does not represent the unknown variable of an augmented matrix). Therefore such a system has no solution.

The uniqueness theorem of linear algebra stipulates that the set of solutions of a system of equations contains a unique solution when there are no free variables. Conversely, the matrix has an infinite number of solutions when there is a least one free variable. Therefore, the presence of free variables determines the uniqueness of the solutions of a matrix.

In a matrix, a free variable appears when the entry directly below and to the right of the pivot above this entry is zero. Usually this leading entry is a pivot but since it is null in this case, the variable the entry is representing becomes “free” to assume any value.

In the figure above, the 3rd pivot should be below and to the right of the 2nd pivot (4). However, since this 3rd entry is actually null (0) the next entry (5) assumes the role of pivot and the variable A3 is said to be free.

Since A3 is a free variable, every possible choice determines a new solution of the system and every solution of the system is determined by a choice of A3. For instance, if we chose A3 to be equal to 5 then A1 would be equal to 1, A2 would be equal to 2 and A4 would yield 4. If we take another value for the free variable A3, say 7, we would get a complete different set of values for A1, A2, and A4. Every time we change the value of the free variable A3 the complete set of solutions changes accordingly. Therefore the solutions of the system completely depends on the value that is assigned to the free variable A3.

The presence of free variables guaranties an infinite number of solutions. Therefore, free variables are the keys to an unlimited manifestation of divine Attributes. The free variable represents a saint that has attained union in the Attributes of God. This type of saint is called wasilan in Arabic He is not confined to a limited number of divine Attributes and his ever-changing spiritual appearance determines the manifestation of the divine Attributes in the other pivots. These pivots represent the saints who manifest a finite number of divine Attributes only. They are still travelers (or *salikan* in Arabic) and each Attribute that becomes manifest through them is determined by the Attribute the “free” saint displays at every moment.

In Islamic spirituality, the mystic poet Rumi (1207-1273) is one of those saints of God who are often mentioned in the discussions of spiritual union. Here are some excerpts from his most famous poems on the subject:

and

and

Shams-I Tabrizi is Rumi’s spiritual master and as such represents the pivot that is higher in spirituality and closer to the Divine Presence. In these verses is displayed the nature of free variables which are infinite and eternally changing. Rumi’s mention of the “outward lowliness of the earth” is the outward form of the pivot that has attained union and that is manifested in the matrix as the number 0 which is the ultimate form of lowliness.

The infinite number of solutions of a matrix is consequently a manifestation of the infinite number of manifestations of divine Attributes a true saint can exhibit. Farid-ud-Din Attar (1145-1221), one of the most important figures of Islamic history, commented on this topic by relating a story from Aba Yazid al Bistami (804-874), another great saint of Islamic history:

Such was the comment of Aba Yazid (q) on the infinite manifestations of divine Attributes through the heart of the saint who has attained union and who is manifested in linear algebra as the pivot appearing as a 0 in the matrix of a system of equation.