Integer matrix

Key takeaways

Integer matrix belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Integer matrix to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Integer matrix from memory before moving on to harder problems.

Reference excerpt

In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.

Examples

( 5 2 6 0 4 7 3 8 5 9 0 4 3 1 0 − 3 9 0 2 1 ) {\displaystyle \left({\begin{array}{cccr}5&2&6&0\\4&7&3&8\\5&9&0&4\\3&1&0&\!\!\!-3\\9&0&2&1\end{array}}\right)} and ( 1 5 0 0 9 2 1 7 3 ) {\displaystyle \left({\begin{array}{ccc}1&5&0\\0&9&2\\1&7&3\end{array}}\right)}

are both examples of integer matrices.

Properties Invertibility of integer matrices is in general more numerically stable than that of non-integer matrices. The determinant of an integer matrix M {\displaystyle M} is itself an integer, and the adjugate matrix of an integer matrix is also integer matrix. Thus, the determinant of an invertible integer matrix is 1 {\displaystyle 1} or − 1 {\displaystyle -1} , and hence where inverses exist they do not become excessively large (see condition number). Theorems from matrix theory that infer properties from determinants thus avoid the traps induced by ill conditioned (nearly zero determinant) real or floating point valued matrices. The inverse of an integer matrix M {\displaystyle M} is again an integer matrix if and only if the determinant of M {\displaystyle M} equals 1 {\displaystyle 1} or − 1 {\displaystyle -1} . Integer matrices of determinant 1 {\displaystyle 1} form the group S L n ( Z ) {\displaystyle \mathrm {SL} _{n}(\mathbf {Z} )} , which has far-reaching applications in arithmetic and geometry. For n = 2 {\displaystyle n=2} , it is closely related to the modular group. The intersection of the integer matrices with the orthogonal group is the group of signed permutation matrices. The characteristic polynomial of an integer matrix has integer coefficients. Since the eigenvalues of a matrix are the roots of this polynomial, the eigenvalues of an integer matrix are algebraic integers. In dimension less than 5, they can thus be expressed by radicals involving integers. Integer matrices are sometimes called integral matrices, although this use is discouraged.

See also GCD matrix Unimodular matrix Wilson matrix

External links Integer Matrix at MathWorld

References

Worked examples

Example 1 — a first encounter with Integer matrix

Start with the simplest possible case. Write down what Integer matrix claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Integer matrix before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Integer matrix ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Integer matrix

In research: Integer matrix appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Integer matrix in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Integer matrix is common in secondary-school and first-year university syllabi. It links to neighbouring topics Matrices (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Integer matrix outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Integer matrix in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Integer matrix means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Integer matrix out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Integer matrix in simple terms?

In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others.

Why does Integer matrix matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Integer matrix?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Integer matrix.