mathematics

Weakly chained diagonally dominant matrix

Weakly chained diagonally dominant 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 Weakly chained diagonally dominant matrix rather than just read about it. In short: In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices. Definition Preliminaries We say row i {\displaystyle i} of a complex matrix A = ( a i j ) {\displaystyle A=(a_{ij})} is strictly diagonally dominant (SDD) if | a i i | > ∑ j ≠ i | a i j | {\displaystyle |a_{ii}|>\textstyle {\sum _{j\neq i}}|a_{ij}|} .

Key takeaways

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

Reference excerpt

In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.

Definition

Preliminaries We say row i {\displaystyle i} of a complex matrix A = ( a i j ) {\displaystyle A=(a_{ij})} is strictly diagonally dominant (SDD) if | a i i | > ∑ j ≠ i | a i j | {\displaystyle |a_{ii}|>\textstyle {\sum _{j\neq i}}|a_{ij}|} . We say A {\displaystyle A} is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with ≥ {\displaystyle \geq } instead. The directed graph associated with an m × m {\displaystyle m\times m} complex matrix A = ( a i j ) {\displaystyle A=(a_{ij})} is given by the vertices { 1 , … , m } {\displaystyle \{1,\ldots ,m\}} and edges defined as follows: there exists an edge from i → j {\displaystyle i\rightarrow j} if and only if a i j ≠ 0 {\displaystyle a_{ij}\neq 0} .

Definition A complex square matrix A {\displaystyle A} is said to be weakly chained diagonally dominant (WCDD) if

A {\displaystyle A} is WDD and for each row i 1 {\displaystyle i_{1}} that is not SDD, there exists a walk i 1 → i 2 → ⋯ → i k {\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}} in the directed graph of A {\displaystyle A} ending at an SDD row i k {\displaystyle i_{k}} .

Example

The m × m {\displaystyle m\times m} matrix

( 1 − 1 1 − 1 1 ⋱ ⋱ − 1 1 ) {\displaystyle {\begin{pmatrix}1\\-1&1\\&-1&1\\&&\ddots &\ddots \\&&&-1&1\end{pmatrix}}}

is WCDD.

Properties

Nonsingularity A WCDD matrix is nonsingular. Proof: Let A = ( a i j ) {\displaystyle A=(a_{ij})} be a WCDD matrix. Suppose there exists a nonzero x {\displaystyle x} in the null space of A {\displaystyle A} . Without loss of generality, let i 1 {\displaystyle i_{1}} be such that | x i 1 | = 1 ≥ | x j | {\displaystyle |x_{i_{1}}|=1\geq |x_{j}|} for all j {\displaystyle j} . Since A {\displaystyle A} is WCDD, we may pick a walk i 1 → i 2 → ⋯ → i k {\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}} ending at an SDD row i k {\displaystyle i_{k}} . Taking moduli on both sides of

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Weakly chained diagonally dominant matrix

Start with the simplest possible case. Write down what Weakly chained diagonally dominant 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 Weakly chained diagonally dominant 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 Weakly chained diagonally dominant 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 Weakly chained diagonally dominant matrix

In research: Weakly chained diagonally dominant 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 Weakly chained diagonally dominant 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: Weakly chained diagonally dominant 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 Weakly chained diagonally dominant 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Weakly chained diagonally dominant matrix” →

Affiliate

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

How to study Weakly chained diagonally dominant matrix in 20 minutes

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

Frequently asked questions

What is Weakly chained diagonally dominant matrix in simple terms?

In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices. Definition Preliminaries We say row i {\displaystyle i} of a complex matrix A = ( a i j ) {\displaystyle A=(a_{ij})} is strictly diagonally do…

Why does Weakly chained diagonally dominant 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 Weakly chained diagonally dominant 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 Weakly chained diagonally dominant matrix.