Samuelson–Berkowitz algorithm

Key takeaways

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

Reference excerpt

In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an n × n {\displaystyle n\times n} matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no divisions, so may be applied to a wider range of algebraic structures.

Description of the algorithm The Samuelson–Berkowitz algorithm applied to a matrix A {\displaystyle A} produces a vector whose entries are the coefficient of the characteristic polynomial of A {\displaystyle A} . It computes this coefficients vector recursively as the product of a Toeplitz matrix and the coefficients vector an ( n − 1 ) × ( n − 1 ) {\displaystyle (n-1)\times (n-1)} principal submatrix. Let A 0 {\displaystyle A_{0}} be an n × n {\displaystyle n\times n} matrix partitioned so that

A 0 = [ a 1 , 1 R C A 1 ] {\displaystyle A_{0}=\left[{\begin{array}{c|c}a_{1,1}&R\\\hline C&A_{1}\end{array}}\right]}

The first principal submatrix of A 0 {\displaystyle A_{0}} is the ( n − 1 ) × ( n − 1 ) {\displaystyle (n-1)\times (n-1)} matrix A 1 {\displaystyle A_{1}} . Associate with A 0 {\displaystyle A_{0}} the ( n + 1 ) × n {\displaystyle (n+1)\times n} Toeplitz matrix T 0 {\displaystyle T_{0}}

defined by

T 0 = [ 1 − a 1 , 1 ] {\displaystyle T_{0}=\left[{\begin{array}{c}1\\-a_{1,1}\end{array}}\right]}

if A 0 {\displaystyle A_{0}} is 1 × 1 {\displaystyle 1\times 1} ,

T 0 = [ 1 0 − a 1 , 1 1 − R C − a 1 , 1 ] {\displaystyle T_{0}=\left[{\begin{array}{c c}1&0\\-a_{1,1}&1\\-RC&-a_{1,1}\end{array}}\right]}

if A 0 {\displaystyle A_{0}} is 2 × 2 {\displaystyle 2\times 2} , and in general

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Samuelson–Berkowitz algorithm

Start with the simplest possible case. Write down what Samuelson–Berkowitz algorithm 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 Samuelson–Berkowitz algorithm 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 Samuelson–Berkowitz algorithm 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 Samuelson–Berkowitz algorithm

In research: Samuelson–Berkowitz algorithm 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 Samuelson–Berkowitz algorithm 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: Samuelson–Berkowitz algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Linear algebra, Numerical linear algebra, Polynomials, so understanding it makes those chapters shorter.
In everyday life: Look for Samuelson–Berkowitz algorithm 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 Samuelson–Berkowitz algorithm in 20 minutes

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

Frequently asked questions

What is Samuelson–Berkowitz algorithm in simple terms?

In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an n × n {\displaystyle n\times n} matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no divisions, so may be applied to a wi…

Why does Samuelson–Berkowitz algorithm 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 Samuelson–Berkowitz algorithm?

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 Samuelson–Berkowitz algorithm.