In mathematics, a **periodic matrix set** is a set of square matrices in which each square matrix is of a different size, and such that each cell within each matrix within a set contains data associated with some type of periodic distribution.

A set may be specified to contain a fixed number of matrices and is identified by a set number (*S*_{M}), where *S* is the set identification number and *M* is the number of matrices included in the set. There is no limit to the number of matrices which may be members of a periodic set.

Each matrix within a set has an identification number (a) and must contain a "root cell". A root cell must be located at any corner of a matrix. All root cells must be located at the same corner of each matrix within a single set. A diagonal line drawn from a root cell to the opposite corner of the same matrix is a "root diagonal".

The periodicity is defined by "partial square rings" (rings) of cells adjoining a root cell on two sides. All cells within the same ring, (even if they are located in a different matrix) have a similar "period". If a matrix contains (n+1)^{2} cells then the outermost ring contains "2n+1" cells which are all included in the same period. A ring identification number (n) identifies each period. The root cell is also the smallest ring and is identified as; n = 0. Each subsequent ring (1, 2, 3, etc.) has 2n+1 cells (3, 5, 7, etc.).

Individual cells contained within a ring are identified by their deviation from the root diagonal. Each cell within a ring is assigned a deviation number (D). All cells intersected by the root diagonal have; D = 0. All cell locations in a column deviation have positive values of D. All cell locations in a row deviation have negative values of D.

Any cell within a set will require three numbers for the identification of its location;

a is the matrix number

n is the ring number

D is the deviation number

The cell could also have its location identified as;

a is the matrix number

x is the column number (root cell = 0)

y is the row number (root cell = 0)

The two locational systems are analogous to Radial (anD) and Cartesian (axy) systems. Generally this article will use the "anD" locational method.

The contents of any cell must contain data that is periodic in some manner.

Combinations of sets are possible; however each set must be conformable for combination. A resultant set (R_{N M}) is the combination of N sets each having M matrices.

Two sets (of compatible construction) may combine so that the root cells on similar sized matrices are adjacent. This is a "set pair" and is identified by a "pair number" (P). The resultant matrices are not square but are 2n x n rectangular.

Four sets may also combine so that all root cells on similar sized matrices are adjacent. This gives a resultant set of square matrices having an even number of cells on each side. All root cells will form a central 2x2 "core" within each resultant matrix. The resultant set is actually two pairs. Each pair forms half of the resultant set. The identifiers (P,S) will tag each quadrant of the resultant set, which is all of the original sets.

P = +½ represents the upper pair

P = -½ represents the lower pair

S = +½ represents the right set of each pair.

S = -½ represents the left set of each pair.

Five identifiers are required to locate any cell in R_{4 M};

P is the pair number

S is the set number

a is the matrix number

n is the ring number

D is the deviation number

Periodic matrix sets have an application to chemistry (for example, in the periodic table) and particle physics (for example, with sub atomic particles). The resultant set R_{4 4} is of special interest.

The periodic rings may be associated with quantum harmonic oscillation. A quantum harmonic oscillator has energy (E_{n}) defined as; E_{n} = (n + ½)ћω. Where; ћ = h/2π and h is Planck's constant, and ω is frequency. The number of cells in each period may be written as; 2E_{n}/ћω.

The rings may also be associated with atomic orbitals. If the ring number (n) is equal to the quantum number for orbital angular momentum (the azimuthal number **l** ), then the rings (0, 1, 2, 3) correspond to the orbitals (s, p, d, f). The ring number is NOT equal to the principal quantum number (**n** ). The number of cells per ring is half the number of electrons per orbital due to spin duality of the electrons.

The quantum numbers are;

**n** is the principal quantum number

**l** is the quantum number for orbital angular momentum (the azimuthal number)

**ml** is the orbital magnetic moment

**ms** is the spin magnetic moment

The spin quantum number (**s**) is not normally used in chemistry applications as all electrons are; **s** = ½.

The atomic number (**Z**) may be expressed as a function of energies which in turn are functions of the quantum numbers.

If a resultant set is R_{4 4} then the locational numbers correspond to the quantum numbers as follows.

S = **ms**

n = **l**

D = **ml**

The Madelung rule gives the "P" and "a" relationships. This rule may be generalized as follows;

2a - P = **n** + **l** + **s**

(2**n** + 2**l** + 2**s** - 4a)^{2} = 1

This generalization may also be obtained from **J** coupling.

If; P = -½

Then; a = ½(**n** + **l**)

If; P = +½

Then; a = ½(**n** + **l**) + ½

The sub-atomic particles may be grouped as an R_{4 3} combination.

A set is considered to be "locationally compliant" if the data contained in each cell is also a function of the location of the cell. Let an R_{4 4} resultant set be populated with atomic numbers. Each cell contains one atomic number (**Z**). The number in each cell should be a function of the locators of the cell. If a term is associated with each locator then the atomic number will be the sum of all terms and a constant.

**Z** = **Z**_{P} + **Z**_{S} + **Z**_{a} + **Z**_{n} + **Z**_{D} - ½

The five locator terms are as follows.

**Z**_{P} = -2a^{2}(P+½)

**Z**_{S} = -2(n+½)(S+½)

**Z**_{a} = 4a(a+1)(a+½)/3

**Z**_{n} = -2n(n+½)

**Z**_{D} = (D+½)

This distribution of atomic numbers in R_{4 4} is a locationally compliant matrix set of the Periodic Table. The following tables show the resultant matrices populated with the atomic numbers.

**R**_{4 4} showing combined matrices 1 to 4 populated with atomic number (Z)

**a = 1**

**a = 2**

**a = 3**

**a = 4**