Identity matrix

Key takeaways

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

Reference excerpt

In linear algebra, the identity matrix of size n {\displaystyle n} is the n × n {\displaystyle n\times n} square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.

Terminology and notation The identity matrix is often denoted by I n {\displaystyle I_{n}} , or simply by I {\displaystyle I} if the size is immaterial or can be trivially determined by the context.

I 1 = [ 1 ] , I 2 = [ 1 0 0 1 ] , I 3 = [ 1 0 0 0 1 0 0 0 1 ] , … , I n = [ 1 0 0 ⋯ 0 0 1 0 ⋯ 0 0 0 1 ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 ] . {\displaystyle I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \dots ,\ I_{n}={\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}.}

The term unit matrix has also been widely used, but the term identity matrix is now standard. The term unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n × n {\displaystyle n\times n} matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, 1 {\displaystyle \mathbf {1} } , or called "id" (short for identity). Less frequently, some mathematics books use U {\displaystyle U} or E {\displaystyle E} to represent the identity matrix, standing for "unit matrix" and the German word Einheitsmatrix respectively. In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as

I n = diag ⁡ ( 1 , 1 , … , 1 ) . {\displaystyle I_{n}=\operatorname {diag} (1,1,\dots ,1).}

The identity matrix can also be written using the Kronecker delta notation:

( I n ) i j = δ i j . {\displaystyle (I_{n})_{ij}=\delta _{ij}.}

Properties When A {\displaystyle A} is an m × n {\displaystyle m\times n} matrix, it is a property of matrix multiplication that

… excerpt ends here. Continue reading the full article.

Illustrations

Identity matrix: Example of a 3x3 identity matrix — Example of a 3x3 identity matrix

Worked examples

Example 1 — a first encounter with Identity matrix

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

In research: Identity 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 Identity 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: Identity matrix is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1 (number), Matrices (mathematics), Sparse matrices, so understanding it makes those chapters shorter.
In everyday life: Look for Identity 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 Identity matrix in 20 minutes

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

Frequently asked questions

What is Identity matrix in simple terms?

In linear algebra, the identity matrix of size n {\displaystyle n} is the n × n {\displaystyle n\times n} square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchan…

Why does Identity 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 Identity 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 Identity matrix.