Productive matrix

Key takeaways

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

Reference excerpt

In linear algebra, a square nonnegative matrix A {\displaystyle A} of order n {\displaystyle n} is said to be productive, or to be a Leontief matrix, if there exists a n × 1 {\displaystyle n\times 1} nonnegative column matrix P {\displaystyle P} such as P − A P {\displaystyle P-AP} is a positive matrix.

History The concept of productive matrix was developed by the economist Wassily Leontief (Nobel Prize in Economics in 1973) in order to model and analyze the relations between the different sectors of an economy. The interdependency linkages between the latter can be examined by the input-output model with empirical data.

Explicit definition The matrix A ∈ M n , n ( R ) {\displaystyle A\in \mathrm {M} _{n,n}(\mathbb {R} )} is productive if and only if A ⩾ 0 {\displaystyle A\geqslant 0} and ∃ P ∈ M n , 1 ( R ) , P > 0 {\displaystyle \exists P\in \mathrm {M} _{n,1}(\mathbb {R} ),P>0} such as P − A P > 0 {\displaystyle P-AP>0} . Here M r , c ( R ) {\displaystyle \mathrm {M} _{r,c}(\mathbb {R} )} denotes the set of r×c matrices of real numbers, whereas > 0 {\displaystyle >0} and ⩾ 0 {\displaystyle \geqslant 0} indicates a positive and a nonnegative matrix, respectively.

Properties The following properties are proven e.g. in the textbook (Michel 1984).

Characterization Theorem A nonnegative matrix A ∈ M n , n ( R ) {\displaystyle A\in \mathrm {M} _{n,n}(\mathbb {R} )} is productive if and only if I n − A {\displaystyle I_{n}-A} is invertible with a nonnegative inverse, where I n {\displaystyle I_{n}} denotes the n × n {\displaystyle n\times n} identity matrix. Proof "If" :

Let I n − A {\displaystyle I_{n}-A} be invertible with a nonnegative inverse, Let U ∈ M n , 1 ( R ) {\displaystyle U\in \mathrm {M} _{n,1}(\mathbb {R} )} be an arbitrary column matrix with U > 0 {\displaystyle U>0} . Then the matrix P = ( I n − A ) − 1 U {\displaystyle P=(I_{n}-A)^{-1}U} is nonnegative since it is the product of two nonnegative matrices. Moreover, P − A P = ( I n − A ) P = ( I n − A ) ( I n − A ) − 1 U = U > 0 {\displaystyle P-AP=(I_{n}-A)P=(I_{n}-A)(I_{n}-A)^{-1}U=U>0} . Therefore A {\displaystyle A} is productive. "Only if" :

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Productive matrix

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

In research: Productive 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 Productive 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: Productive matrix is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra, Mathematical economics, Matrices (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Productive 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 Productive matrix in 20 minutes

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

Frequently asked questions

What is Productive matrix in simple terms?

In linear algebra, a square nonnegative matrix A {\displaystyle A} of order n {\displaystyle n} is said to be productive, or to be a Leontief matrix, if there exists a n × 1 {\displaystyle n\times 1} nonnegative column matrix P {\displaystyle P} such as P − A P {\displaystyle P-AP} is a positive ma…

Why does Productive 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 Productive 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 Productive matrix.