Nullity theorem

Key takeaways

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

Reference excerpt

The nullity theorem is a mathematical theorem about the inverse of a partitioned matrix, which states that the nullity of a block in a matrix equals the nullity of the complementary block in its inverse matrix. Here, the nullity is the dimension of the kernel. The theorem was proven in an abstract setting by Gustafson (1984), and for matrices by (Fiedler & Markham 1986). Partition a matrix and its inverse in four submatrices:

[ A B C D ] − 1 = [ E F G H ] . {\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}={\begin{bmatrix}E&F\\G&H\end{bmatrix}}.}

The partition on the right-hand side should be the transpose of the partition on the left-hand side, in the sense that if A is an m-by-n block then E should be an n-by-m block. The statement of the nullity theorem is now that the nullities of the blocks on the right equal the nullities of the blocks on the left (Strang & Nguyen 2004):

nullity A = nullity H , nullity B = nullity F , nullity C = nullity G , nullity D = nullity E . {\displaystyle {\begin{aligned}\operatorname {nullity} \,A&=\operatorname {nullity} \,H,\\\operatorname {nullity} \,B&=\operatorname {nullity} \,F,\\\operatorname {nullity} \,C&=\operatorname {nullity} \,G,\\\operatorname {nullity} \,D&=\operatorname {nullity} \,E.\end{aligned}}}

More generally, if a submatrix is formed from the rows with indices {i1, i2, …, im} and the columns with indices {j1, j2, …, jn}, then the complementary submatrix is formed from the rows with indices {1, 2, …, N} \ {j1, j2, …, jn} and the columns with indices {1, 2, …, N} \ {i1, i2, …, im}, where N is the size of the whole matrix. The nullity theorem states that the nullity of any submatrix equals the nullity of the complementary submatrix of the inverse.

References Gustafson, William H. (1984), "A note on matrix inversion", Linear Algebra and Its Applications, 57: 71–73, doi:10.1016/0024-3795(84)90177-0, ISSN 0024-3795. Fiedler, Miroslav; Markham, Thomas L. (1986), "Completing a matrix when certain entries of its inverse are specified", Linear Algebra and Its Applications, 74 (1–3): 225–237, doi:10.1016/0024-3795(86)90125-4, ISSN 0024-3795. Strang, Gilbert; Nguyen, Tri (2004), "The interplay of ranks of submatrices" (PDF), SIAM Review, 46 (4): 637–646, Bibcode:2004SIAMR..46..637S, doi:10.1137/S0036144503434381, hdl:1721.1/3885, ISSN 1095-7200.

Worked examples

Example 1 — a first encounter with Nullity theorem

Start with the simplest possible case. Write down what Nullity theorem 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 Nullity theorem 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 Nullity theorem 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 Nullity theorem

In research: Nullity theorem 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 Nullity theorem 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: Nullity theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Matrix theory, Theorems in linear algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Nullity theorem 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 Nullity theorem in 20 minutes

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

Frequently asked questions

What is Nullity theorem in simple terms?

The nullity theorem is a mathematical theorem about the inverse of a partitioned matrix, which states that the nullity of a block in a matrix equals the nullity of the complementary block in its inverse matrix. Here, the nullity is the dimension of the kernel.

Why does Nullity theorem 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 Nullity theorem?

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 Nullity theorem.