P-matrix

Key takeaways

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

Reference excerpt

In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P 0 {\displaystyle P_{0}} -matrices, which are the closure of the class of P-matrices, with every principal minor ≥ {\displaystyle \geq } 0.

Spectra of P-matrices By a theorem of Kellogg, the eigenvalues of P- and P 0 {\displaystyle P_{0}} - matrices are bounded away from a wedge about the negative real axis as follows:

If { u 1 , . . . , u n } {\displaystyle \{u_{1},...,u_{n}\}} are the eigenvalues of an n-dimensional P-matrix, where n > 1 {\displaystyle n>1} , then

| arg ⁡ ( u i ) | < π − π n , i = 1 , . . . , n {\displaystyle |\arg(u_{i})|<\pi -{\frac {\pi }{n}},\ i=1,...,n}

If { u 1 , . . . , u n } {\displaystyle \{u_{1},...,u_{n}\}} , u i ≠ 0 {\displaystyle u_{i}\neq 0} , i = 1 , . . . , n {\displaystyle i=1,...,n} are the eigenvalues of an n-dimensional P 0 {\displaystyle P_{0}} -matrix, then

| arg ⁡ ( u i ) | ≤ π − π n , i = 1 , . . . , n {\displaystyle |\arg(u_{i})|\leq \pi -{\frac {\pi }{n}},\ i=1,...,n}

Remarks The class of nonsingular M-matrices is a subset of the class of P-matrices. More precisely, all matrices that are both P-matrices and Z-matrices are nonsingular M-matrices. The class of sufficient matrices is another generalization of P-matrices. The linear complementarity problem L C P ( M , q ) {\displaystyle \mathrm {LCP} (M,q)} has a unique solution for every vector q if and only if M is a P-matrix. This implies that if M is a P-matrix, then M is a Q-matrix. If the Jacobian of a function is a P-matrix, then the function is injective on any rectangular region of R n {\displaystyle \mathbb {R} ^{n}} . A related class of interest, particularly with reference to stability, is that of P ( − ) {\displaystyle P^{(-)}} -matrices, sometimes also referred to as N − P {\displaystyle N-P} -matrices. A matrix A is a P ( − ) {\displaystyle P^{(-)}} -matrix if and only if ( − A ) {\displaystyle (-A)} is a P-matrix (similarly for P 0 {\displaystyle P_{0}} -matrices). Since σ ( A ) = − σ ( − A ) {\displaystyle \sigma (A)=-\sigma (-A)} , the eigenvalues of these matrices are bounded away from the positive real axis.

See also Routh–Hurwitz matrix Linear complementarity problem M-matrix Q-matrix Z-matrix Perron–Frobenius theorem

Notes

References Csizmadia, Zsolt; Illés, Tibor (2006). "New criss-cross type algorithms for linear complementarity problems with sufficient matrices" (PDF). Optimization Methods and Software. 21 (2): 247–266. doi:10.1080/10556780500095009. MR 2195759. David Gale and Hukukane Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965) doi:10.1007/BF01360282 Li Fang, On the Spectra of P- and P 0 {\displaystyle P_{0}} -Matrices, Linear Algebra and its Applications 119:1-25 (1989) R. B. Kellogg, On complex eigenvalues of M and P matrices, Numer. Math. 19:170-175 (1972)

Worked examples

Example 1 — a first encounter with P-matrix

Start with the simplest possible case. Write down what P-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 P-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 P-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 P-matrix

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

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

Frequently asked questions

What is P-matrix in simple terms?

In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P 0 {\displaystyle P_{0}} -matrices, which are the closure of the class of P-matrices, with every principal minor ≥ {\displaystyle \geq } 0.

Why does P-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 P-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 P-matrix.