Can an identity matrix be rectangular?

The standard multiplicative identity I_n is square. Some software creates rectangular diagonal arrays, but they do not serve as a two-sided identity on one vector space.

What is the inverse of I_n?

It is I_n itself, because I_n multiplied by itself remains I_n.

Generally no. Adding I increases the diagonal entries by one when the dimensions match.

AI Glossary / Mathematics and optimization / Beginner

Identity matrix

Definition

An identity matrix is a square matrix with ones on its main diagonal and zeros everywhere else. Multiplying a compatible vector or matrix by it leaves that object unchanged. Written I, or I_n to specify n rows and n columns, it plays the role in matrix multiplication that the number one plays in ordinary multiplication.

What it means

A matrix can describe a transformation of a vector: stretching, rotating, mixing, or otherwise combining coordinates. The identity transformation returns every input vector exactly as it was. Its matrix has one direct connection from each input coordinate to the corresponding output coordinate and no mixing between different coordinates.
The size matters. I_2 acts on two-component column vectors, while I_3 acts on three-component ones. For a matrix A with m rows and n columns, the compatible identities satisfy I_m A=A and A I_n=A. The left and right identity matrices need not have the same size when A is rectangular.

How it works

Matrix multiplication forms each output entry by multiplying corresponding row and column entries and adding the products. In the identity matrix, only one entry of each row is nonzero. Consequently, the multiplication selects the matching input coordinate with coefficient one.
The columns of I_n are the standard coordinate vectors. They are also a useful reference for transformations: the jth column of A is the result of applying A to the jth standard vector. Identity matrices appear in equations for inverses, unchanged mappings, and shifts such as A+lambda I. Multiplication by I preserves A, whereas adding I generally changes it.

Three practical examples

A graphics or robotics calculation can use an identity transformation as its unchanged starting transformation before composing movements.
A numerical algorithm can compare a computed inverse product with the identity to inspect whether the inverse relationship is approximately satisfied.
A machine-learning calculation can add a scaled identity to an appropriate square matrix when defining a regularized linear system.

Think of it this way

Think of a switchboard that sends each channel straight to its matching output at unchanged strength. No channel is mixed with another. The analogy is limited to the mathematical mapping: an actual physical switchboard can introduce noise or delay, whereas exact multiplication by an identity matrix introduces neither.

Worked example

Take the original vector x=(4,−2,7). The three rows of I_3 are (1,0,0), (0,1,0), and (0,0,1). Their dot products with x are 4, −2, and 7, so I_3x=x.
Now let A be a two-by-three matrix with rows (2,−1,0) and (0,3,5). Multiplying A by x produces (10,29). Multiplying on the right by I_3 leaves all three columns of A unchanged, so (A I_3)x also gives (10,29). Multiplying A on the left by I_2 leaves its two rows unchanged. Attempting A I_2 is not a valid matrix product because the inner dimensions, three and two, do not match.
For a separate square example, let B have rows (2,1) and (0,3). Then B+2I_2 has rows (4,1) and (0,5): only diagonal entries increase. By contrast, adding two to every entry would produce rows (4,3) and (2,5), a different matrix.
Finally, a candidate inverse C with rows (1/2,−1/6) and (0,1/3) satisfies BC=I_2. The upper-right product is 2(−1/6)+1(1/3)=0; the diagonal products are one. Checking CB gives I_2 as well. These calculations illustrate the identity's role in an inverse equation without suggesting that explicitly computing inverses is always the best way to solve linear systems.

Illustrative case study

In a fictional example, ClearFrame builds a small coordinate-processing tool. A developer initializes a transformation with a matrix filled entirely with ones, expecting it to leave points unchanged. Testing the vector (2,5) instead produces (7,7), revealing that both output coordinates sum the inputs.
Replacing that matrix with I_2 returns (2,5), as intended. The team adds a simple unchanged-transformation check before composing other transformations. This invented case illustrates the difference between an identity and an all-ones matrix; it does not describe an actual product defect.

Common misconceptions and mistakes

An identity matrix is not an all-ones matrix. For dimensions larger than one, the off-diagonal zeros are essential to prevent coordinate mixing. It is also not a zero matrix: a zero matrix sends every compatible vector to zero.
Matrix multiplication and elementwise multiplication are different operations. Elementwise multiplication of a square A by I keeps its diagonal and removes its off-diagonal entries; it generally does not return A.
Do not infer statistical independence merely because a covariance matrix is the identity. Identity covariance means unit variances and zero pairwise covariances. Independence requires additional assumptions, such as a jointly Gaussian model, or separate justification.

Business use

The identity matrix is foundational rather than a standalone business technique. Understanding it helps teams interpret model formulas, transformation pipelines, and numerical diagnostics accurately. It is particularly useful when checking dimensions or explaining why a parameter changes only diagonal terms.
In implementation reviews, specify whether an operation means matrix multiplication, elementwise multiplication, or addition. Also avoid constructing a large dense identity merely to perform a conceptual no-op; numerical software can often represent or apply the operation without storing every zero. The mathematical identity is exact, while numerical checks involving computed matrices should allow an appropriate floating-point tolerance.

Check your understanding

A has five rows and two columns. Which identity matrices preserve it under left and right multiplication?

Answer: I_5 A=A and A I_2=A. Their sizes match the row and column dimensions respectively.

Related terms

Matrix
Inverse matrix
Regularization
Covariance matrix
identity transformation
diagonal matrix
standard basis
matrix multiplication

Learn next

Choose a small rectangular matrix and verify both compatible identity products. Then compare matrix multiplication by I with elementwise multiplication by I for a square matrix containing off-diagonal entries. Explain why only the first operation is an unchanged transformation.

Matrix

Sources

Primary teaching source: Gilbert Strang, Introduction to Linear Algebra, section 2.5, Inverse Matrices: https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_2-5.pdf . Identity and inverse relationships were consulted. The numerical examples and fictional implementation case are original.

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