Identity Matrix

Right. Let's get this over with. Before you get lost in the labyrinth of linear algebra, you'll need to understand its most fundamental, and frankly, most boring component. Pay attention; I won't be repeating myself.

And for the record, try to keep up. This is not to be confused with a matrix of ones, a concept for which the name is tragically literal. Nor is it a unitary matrix or a matrix unit. If you can't distinguish between doing nothing and doing something, we have more significant problems to address.

In the field of linear algebra, the identity matrix of size n is the n × n square matrix defined by the minimalist principle of placing ones on the main diagonal and zeros in every other tedious position. Its defining characteristic is its profound lack of impact. When it represents a geometric transformation, it is the mathematical equivalent of a shrug; the object remains utterly unchanged. In other contexts, it's the matrix version of multiplying by the number 1—a necessary, yet deeply uninspired, function.

Terminology and notation

The identity matrix is most commonly denoted by I_n, or simply I if the size is so painfully obvious from the context that specifying it would be an insult to everyone's intelligence.

Here they are, in all their underwhelming glory:

${\displaystyle I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \dots ,\ I_{n}={\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}.}$

The term "unit matrix" has also seen some use, a historical artifact now largely abandoned by anyone with a preference for clarity. While you might find it in older texts, the term "identity matrix" is now standard for a reason. "Unit matrix" is an ambiguous phrase, as it's also been used to describe a matrix of ones (a matrix filled with nothing but ones) and, more broadly, any unit within the ring of all n × n matrices. Precision is not optional here.

In certain specialized corners of the universe, such as group theory or quantum mechanics, the identity matrix is sometimes represented by a boldface one, 1, or called "id," a shorthand for identity. Less frequently, you might see older mathematics books use U or E, representing "unit matrix" or the German Einheitsmatrix, respectively. Quaint affectations, but you've been warned.

For those who prefer a notation that doesn't waste ink, the identity matrix can be concisely described as a diagonal matrix:

${\displaystyle I_{n}=\operatorname {diag} (1,1,\dots ,1).}$

It can also be expressed with the pristine elegance of the Kronecker delta notation, which is arguably the only truly civilized way to write it:

${\displaystyle (I_{n})_{ij}=\delta _{ij}.}$

This simply states that the element at row i and column j is 1 if i equals j, and 0 otherwise. Simple. Clean. Final.

Properties

Let A be an m × n matrix. It is a fundamental property of matrix multiplication that the identity matrix leaves A entirely unscathed:

${\displaystyle I_{m}A=AI_{n}=A.}$

This makes the identity matrix the multiplicative identity within the matrix ring of all n × n matrices. It is also the identity element of the general linear group GL(n), which is the set of all invertible n × n matrices. The identity matrix is, of course, invertible. It is its own inverse, a perfect loop of self-sufficiency. This makes it an involutory matrix. Within this group, if the product of two square matrices is the identity matrix, it means they are, by definition, inverses of each other. They cancel each other out, returning to the state of perfect neutrality that is the identity.

When n × n matrices are employed to represent linear transformations from an n-dimensional vector space back to itself, the identity matrix I_n represents the identity function—the transformation of doing absolutely nothing. An object subjected to this transformation remains exactly where it started, regardless of the basis used for the representation. It is the embodiment of inertia.

Structurally, the i-th column of an identity matrix is the unit vector e_i. This is a vector whose i-th entry is 1, with all other entries being 0. It's the building block of the standard basis, and the identity matrix is just these blocks lined up in perfect, unimaginative order.

The determinant of the identity matrix is 1. This signifies that the transformation it represents preserves volume and orientation. Of course it does; it doesn't do anything. Its trace—the sum of the elements on its main diagonal—is simply n, the dimension of the space it inhabits.

Here’s something marginally more interesting: the identity matrix is the only idempotent matrix with a non-zero determinant. To be idempotent means that when a matrix is multiplied by itself, it yields itself. Most idempotent matrices are projections, collapsing space onto a smaller subspace and thereby having a determinant of zero. The identity matrix is the only one that satisfies two conditions simultaneously:

• When multiplied by itself, the result is itself (I * I = I).
• All of its rows and columns are linearly independent.

This combination makes it uniquely stable. It projects the entire vector space perfectly back onto itself without any loss of dimension or information.

The principal square root of an identity matrix is, unsurprisingly, itself. This is also its only positive-definite square root. However, and here lies a flicker of complexity in the otherwise placid creature, every identity matrix with at least two rows and columns possesses an infinitude of symmetric square roots. A quiet chaos of possibilities hiding beneath a surface of absolute order.

Finally, the rank of an identity matrix I_n is equal to its size, n.

${\displaystyle \operatorname {rank} (I_{n})=n.}$

It has full rank, meaning its column vectors span the entire n-dimensional space. It may be passive, but it is undeniably present across every dimension.

See also

• Binary matrix (zero-one matrix)
• Elementary matrix
• Exchange matrix
• Matrix of ones
• Pauli matrices (the identity matrix is the zeroth Pauli matrix)
• Householder transformation (the Householder matrix is built through the identity matrix)
• Square root of a 2 by 2 identity matrix
• Unitary matrix
• Zero matrix

Notes

1. ^ "Identity matrix: intro to identity matrices (article)". Khan Academy. Retrieved 2020-08-14.
2. ^ ^{a b} Pipes, Louis Albert (1963). Matrix Methods for Engineering. Prentice-Hall International Series in Applied Mathematics. Prentice-Hall. p. 91.
3. ^ Roger Godement, Algebra, 1968.
4. ^ ISO 80000-2:2009.
5. ^ Ken Stroud, Engineering Mathematics, 2013.
6. ^ ISO 80000-2:2019.
7. ^ Weisstein, Eric W. "Unit Matrix". mathworld.wolfram.com. Retrieved 2021-05-05.
8. ^ ^{a b} Weisstein, Eric W. "Identity Matrix". mathworld.wolfram.com. Retrieved 2020-08-14.
9. ^ Mitchell, Douglas W. (November 2003). "87.57 Using Pythagorean triples to generate square roots of ${\displaystyle I_{2}}$". The Mathematical Gazette. 87 (510): 499–500. doi:10.1017/S0025557200173723. JSTOR 3621289.