mathematics

Tutte matrix

Tutte matrix is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tutte matrix rather than just read about it. In short: In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n-by-n skew-symmetric matrix A with entries A i j = { x i j if ( i , j ) ∈ E and i < j − x j i if ( i , j ) ∈…

Key takeaways

Tutte 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 Tutte matrix to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tutte matrix from memory before moving on to harder problems.

Reference excerpt

In graph theory, the Tutte matrix A of a graph G = (V, E) is a matrix used to determine the existence of a perfect matching: that is, a set of edges which is incident with each vertex exactly once. If the set of vertices is V = { 1 , 2 , … , n } {\displaystyle V=\{1,2,\dots ,n\}} then the Tutte matrix is an n-by-n skew-symmetric matrix A with entries

A i j = { x i j if ( i , j ) ∈ E and i < j − x j i if ( i , j ) ∈ E and i > j 0 otherwise {\displaystyle A_{ij}={\begin{cases}x_{ij}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i<j\\-x_{ji}\;\;{\mbox{if}}\;(i,j)\in E{\mbox{ and }}i>j\\0\;\;\;\;{\mbox{otherwise}}\end{cases}}}

where the xij are indeterminates. The determinant of this matrix is then a polynomial (in the variables xij, i < j ): this coincides with the square of the Pfaffian of the matrix A and is non-zero (as a polynomial) if and only if a perfect matching exists. (This polynomial is not the Tutte polynomial of G.) The Tutte matrix is named after W. T. Tutte, and is a generalisation of the Edmonds matrix for a balanced bipartite graph.

References R. Motwani, P. Raghavan (1995). Randomized Algorithms. Cambridge University Press. p. 167. Allen B. Tucker (2004). Computer Science Handbook. CRC Press. p. 12.19. ISBN 1-58488-360-X. W.T. Tutte (April 1947). "The factorization of linear graphs" (PDF). J. London Math. Soc. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. Retrieved 2008-06-15.

Worked examples

Example 1 — a first encounter with Tutte matrix

Start with the simplest possible case. Write down what Tutte 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 Tutte 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 Tutte 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 Tutte matrix

In research: Tutte 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 Tutte 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: Tutte matrix is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic graph theory, Graph theory stubs, Matching (graph theory), so understanding it makes those chapters shorter.
In everyday life: Look for Tutte 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Tutte matrix in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tutte 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 Tutte matrix out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Tutte matrix in simple terms?

Why does Tutte 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 Tutte 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 Tutte matrix.