mathematics

G-matrix

G-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 G-matrix rather than just read about it. In short: In linear algebra, a real invertible matrix A {\displaystyle A} is called a G-matrix if A − T = D 1 A D 2 {\displaystyle A^{-T}=D_{1}AD_{2}} (where A − T {\displaystyle A^{-T}} means ( A − 1 ) T {\displaystyle (A^{-1})^{T}} ) for some real diagonal matrices D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} . The term "G-matrix" was coined by Miroslav Fiedler and Frank J.

Key takeaways

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

Reference excerpt

In linear algebra, a real invertible matrix A {\displaystyle A} is called a G-matrix if A − T = D 1 A D 2 {\displaystyle A^{-T}=D_{1}AD_{2}} (where A − T {\displaystyle A^{-T}} means ( A − 1 ) T {\displaystyle (A^{-1})^{T}} ) for some real diagonal matrices D 1 {\displaystyle D_{1}} and D 2 {\displaystyle D_{2}} . The term "G-matrix" was coined by Miroslav Fiedler and Frank J. Hall. It is sometimes called a semi-orthogonal matrix in research literature, although the latter term may also refer to a different kind of matrix, namely, a non-square matrix with orthogonal columns/rows. All real orthogonal matrices and real invertible diagonal matrices, for instances, are G-matrices.

Properties All matrices below are assumed to be real square matrices.

If A {\displaystyle A} is a G-matrix, so are A T {\displaystyle A^{T}} and A − 1 {\displaystyle A^{-1}} . If A {\displaystyle A} is a G-matrix and D {\displaystyle D} is a nonsingular diagonal matrix, then both A D {\displaystyle AD} and D A {\displaystyle DA} are G-matrices. If A {\displaystyle A} is a G-matrix and P {\displaystyle P} is a permutation matrix, then both A P {\displaystyle AP} and P A {\displaystyle PA} are G-matrices. If A {\displaystyle A} is a G-matrix, then A {\displaystyle A} and A − T {\displaystyle A^{-T}} have the same entrywise zero pattern, i.e., A i j = 0 {\displaystyle A_{ij}=0} if and only if ( A − T ) i j = 0 {\displaystyle (A^{-T})_{ij}=0} . Thus the entrywise zero patterns of A {\displaystyle A} and A − 1 {\displaystyle A^{-1}} are symmetric to each other. The direct sum of G-matrices is again a G-matrix. Compound matrices of a G-matrix are G-matrices. Kronecker products of G-matrices are G-matrices. Every nonsingular Cauchy matrix C {\displaystyle C} such that C − 1 e {\displaystyle C^{-1}e} and C − T e {\displaystyle C^{-T}e} are entrywise nonzero is a G-matrix. Here e {\displaystyle e} denotes the vector of ones.

References

Worked examples

Example 1 — a first encounter with G-matrix

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

In research: G-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 G-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: G-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 G-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 G-matrix in 20 minutes

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

Frequently asked questions

What is G-matrix in simple terms?

Why does G-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 G-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 G-matrix.