Hellinger–Toeplitz theorem

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In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N which is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b are both integer, and M is unimodular, has an integer solution. The unimodular matrices of order n form a group, which is denoted $GL_{n}(\mathbb {Z} )$.

Examples of unimodular matrices

Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular:

• Identity matrix
• The inverse of a unimodular matrix
• The product of two unimodular matrices

Further:

• The Kronecker product of two unimodular matrices is also unimodular. This follows since

$\det(A\otimes B)=(\det A)^{q}(\det B)^{p},$

where p and q are the dimensions of A and B, respectively.

Concrete examples include:

• Symplectic matrices
• Pascal matrices
• Permutation matrices
• the three transformation matrices in the ternary tree of primitive Pythagorean triples

Total unimodularity

A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A totally unimodular matrix need not be square itself. From the definition it follows that any totally unimodular matrix has only 0, +1 or −1 entries. The opposite is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular.

Totally unimodular matrices are extremely important in polyhedral combinatorics and combinatorial optimization since they give a quick way to verify that a linear program is integral (has an integral optimum, when any optimum exists). Specifically, if A is TU and b is integral, then linear programs of forms like $\{\min cx\mid Ax\geq b,x\geq 0\}$ or $\{\max cx\mid Ax\leq b\}$ have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region (e.g. $\{x\mid Ax\geq b\}$) is integral and thus the feasible region is an integral polyhedron.

Common totally unimodular matrices

1. The unoriented incidence matrix of a bipartite graph, which is the coefficient matrix for bipartite matching, is totally unimodular (TU). (The unoriented incidence matrix of a non-bipartite graph is not TU.) More generally, in the appendix to a paper by Heller and Tompkins, A.J. Hoffman and D. Gale prove the following. Let $A$ be an m by n matrix whose rows can be partitioned into two disjoint sets $B$ and $C$. Then the following four conditions together are sufficient for A to be totally unimodular:

• Every column of $A$ contains at most two non-zero entries;
• Every entry in $A$ is 0, +1, or −1;
• If two non-zero entries in a column of $A$ have the same sign, then the row of one is in $B$, and the other in $C$;
• If two non-zero entries in a column of $A$ have opposite signs, then the rows of both are in $B$, or both in $C$.

It was realized later that these conditions define an incidence matrix of a balanced signed graph; thus, this example says that the incidence matrix of a signed graph is totally unimodular if the signed graph is balanced. The converse is valid for signed graphs without half edges (this generalizes the property of the unoriented incidence matrix of a graph).

2. The constraints of maximum flow and minimum cost flow problems yield a coefficient matrix with these properties (and with empty C). Thus, such network flow problems with bounded integer capacities have an integral optimal value. Note that this does not apply to multi-commodity flow problems, in which it is possible to have fractional optimal value even with bounded integer capacities.

3. The consecutive-ones property: if A is (or can be permuted into) a 0-1 matrix in which for every row, the 1s appear consecutively, then A is TU. (The same holds for columns since the transpose of a TU matrix is also TU.)

4. Every network matrix is TU. The rows of a network matrix correspond to a tree T=(V,R), each of whose arcs has an arbitrary orientation (it is not necessary that there exist a root vertex r such that the tree is "rooted into r" or "out of r").The columns correspond to another set C of arcs on the same vertex set V. To compute the entry at row R and column C=st, look at the s-to-t path P in T; then the entry is:

• +1 if arc R appears forward in P,
• -1 if arc R appears backwards in P,
• 0 if arc R does not appear in P.

See more in Schrijver (2003).

5. Ghouila-Houri showed that a matrix is TU iff for every subset R of rows, there is an assignment $s:R\to \pm 1$ of signs to rows so that the signed sum $\sum _{r\in R}s(r)r$ (which is a row vector of the same width as the matrix) has all its entries in $\{0,\pm 1\}$ (i.e. the row-submatrix has discrepancy at most one). This and several other if-and-only-if characterizations are proven in Schrijver (2003).

6. Hoffman and Kruskal proved the following theorem. Suppose $G$ is a directed graph without any 2-dicycle, $P$ is the set of all dipaths in $G$, and $A$ is the 0-1 incidence matrix of $V(G)$ versus $P$. Then $A$ is totally unimodular if and only if every simple arbitrarily-oriented cycle in $G$ consists of alternating forwards and backwards arcs.

7. Suppose a matrix has 0-($\pm$1) entries and in each column, the entries are non-decreasing from top to bottom (so all -1's are on top, then 0's, then 1's are on the bottom). Fujishige showed that the matrix is TU iff every 2-by-2 submatrix has determinant in $0,\pm 1$.

8. Seymour (1980) proved a full characterization of all TU matrices, which we describe here only informally. Seymour's theorem is that a matrix is TU if and only if it is a certain natural combination of some network matrices and some copies of a particular 5-by-5 TU matrix.

Concrete examples

1. The following matrix is totally unimodular:

$A={\begin{bmatrix}-1&-1&0&0&0&+1\\+1&0&-1&-1&0&0\\0&+1&+1&0&-1&0\\0&0&0&+1&+1&-1\\\end{bmatrix}}.$

This matrix arises as the coefficient matrix of the constraints in the linear programming formulation of the maximum flow problem on the following network:

File:Graph for example adjacency matrix.svg

2. Any matrix of the form

$A={\begin{bmatrix}\vdots &\vdots &\vdots &\vdots &\vdots \\\dotsb &+1&\dotsb &+1&\dotsb \\\vdots &\vdots &\vdots &\vdots &\vdots \\\dotsb &+1&\dotsb &-1&\dotsb \\\vdots &\vdots &\vdots &\vdots &\vdots \\\end{bmatrix}}.$

is not totally unimodular, since it has a square submatrix of determinant -2.

Abstract linear algebra

Abstract linear algebra considers matrices with entries from any commutative ring, not limited to the integers. In this context, a unimodular matrix is one that is invertible over the ring; equivalently, whose determinant is a unit. This group is denoted $GL_{n}R\,$.

Over a field, unimodular has the same meaning as non-singular. Unimodular here refers to matrices with coefficients in some ring (often the integers) which are invertible over that ring, and one uses non-singular to mean matrices that are invertible over the field.

See also

• Balanced matrix
• Regular matroid
• Special linear group
• Total dual integrality

References

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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Alexander Schrijver (1998), Theory of Linear and Integer Programming. John Wiley & Sons, ISBN 0-471-98232-6 (mathematical)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Mathematical Programming Glossary by Harvey J. Greenberg
• Unimodular Matrix from MathWorld
• Software for testing total unimodularity by M. Walter and K. Truemper