Identity Matrix

Introduction

The identity matrix, also called the unit matrix, is a fundamental concept in linear algebra. It is a special type of square matrix that acts as the multiplicative identity in matrix operations, meaning that multiplying any matrix by it leaves the original matrix unchanged. This property makes it essential in various mathematical and practical applications.

The identity matrix plays a key role in solving linear systems, where equations of the form

(Ax = b)

simplify to ( x = b ) when ( A = I ). In eigenvalue problems, every vector is an eigenvector of the identity matrix, and all eigenvalues are equal to 1. It is also widely used in machine learning for regularization techniques such as ridge regression.

Beyond pure mathematics, the identity matrix has practical significance in fields like computer graphics, where it represents a transformation that leaves objects unchanged, and quantum mechanics, where it serves as the identity operator in quantum state calculations.

As a foundational element of matrix algebra, the identity matrix is essential for understanding linear transformations, solving equations, and performing various computations across multiple disciplines.

Definition and Notation

The identity matrix, also known as the unit matrix, is a special type of square matrix in which all the elements on main diagonal are 1, and all other elements are 0.

Mathematically, the identity matrix

I_n

is denoted as

(I_n)_{ij} = e_{ij} = \begin{cases} 1, & \text{if } i = j \\ 0, & \text{if } i \neq j \end{cases}

In matrix algebra we use

I_n

to denote the identity matrix where

n

is the order (number of rows or columns).

The

2 \times 2

identity matrix is

I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

The

3 \times 3

identity matrix is

I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

For a general

n \times n

case, the identity matrix is

I_n = \begin{pmatrix} 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{pmatrix}

Sometimes in cases where the order is clear we can drop the

n

and denote it just as

I

, like here:

I=

\begin{pmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}

This is perfectly valid notation as well for identity matrix.

Basic Properties

Due to it's special structure we discussed in previous section , identity matrix has several funndamental properties:

1. Square Matrix: The identity matrix is always a square matrix of order

n \times n

2. Main Diagonal: All elements on the main diagonal are 1, and all off-diagonal elements are 0.

3. Multiplicative Identity: For any square

n \times n

matrix

A

, multiplying by the identity matrix does not change

A

A I_n = I_n A = A

4. Symmetric Matrix: The identity matrix is always symmetric, meaning

I_n^T = I_n

5. Determinant: The determinant of the identity matrix is always 1, i.e.,

\det(I_n) = 1

6. Trace: The trace (sum of diagonal elements) of an

n \times n

identity matrix is

n

, i.e.,

\text{tr}(I_n) = n

7. Inverse: The identity matrix is its own inverse, meaning

I_n^{-1} = I_n

Closure Properties

The identity matrix follows certain closure properties under different operations.

1. Closed under Multiplication: Multiplying two identity matrices of the same size results in another identity matrix:

I_n \cdot I_n = I_n

2. Closed under Exponentiation: Raising the identity matrix to any integer power always gives the identity matrix:

I_n^k = I_n

, for any integer

k

.
Check here for more about powers of unit matrix

3. Not Closed under Addition: Adding two identity matrices results in a scalar multiple of the identity matrix, not an identity matrix itself:

I_n + I_n = 2I_n

4. Not Closed under Scalar Multiplication: Multiplying the identity matrix by any scalar other than 1 results in a scalar matrix, not an identity matrix:

c I_n

is not an identity matrix for

c \neq 1

Multiplicative Role in Matrix Algebra

The identity matrix is called the unit matrix because it behaves as the multiplicative identity in matrix algebra. This means that when multiplied by another matrix, it leaves the matrix unchanged.

Multiplication with Square Matrices

For any square matrix

A

of size

n \ imes n

, multiplying by the identity matrix does not change

A

Multiplication from the left:

I_n A = A

Multiplication from the right:

A I_n = A

Since matrix multiplication is generally not commutative, the fact that

I_n

satisfies both

I_n A = A

and

A I_n = A

reinforces its role as the identity element in matrix algebra.

Multiplication with Non-Square Matrices

When the identity matrix is multiplied with a non-square matrix, it still acts as the identity, but the order of the identity matrix must match the matrix it is multiplied with.

For a matrix

B

of size

m \times n

, we use identity matrices of appropriate size:

Multiplication from the left: If

B

m \times n

, then

I_m B = B

, where

I_m

is the

m \times m

identity matrix.

Multiplication from the right: If

B

m \times n

, then

B I_n = B

, where

I_n

is the

n \times n

identity matrix.

This means that identity matrices of different sizes still preserve the structure of the matrix they are multiplied with, maintaining their role as unit matrices in different dimensions.

Powers of the Identity Matrix

Exponentiation of matrices is a special case of repeated matrix multiplication. Since the identity matrix is the multiplicative identity, raising it to any integer power always results in itself.

Definition of Matrix Exponentiation

For any square matrix

A

, exponentiation is defined as repeated multiplication:

A^k = A \cdot A \cdots A

(k times). Applying this to the identity matrix:

I_n^k = I_n \cdot I_n \cdots I_n

(k times).

Clearly we can see raising matrices to power as a special case of multiplication.And as we already discussed the multiplication role of identity matrix in previous section, it is obvious that exponentiation leaves it unchanged.

Example: Powers Identity Matrix for

2 \times 2

Consider the identity matrix of order 2:

I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Now, multiplying it by itself:

I_2^2 = I_2 \cdot I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I_2

Similarly, for a higher power:

I_2^3 = I_2 \cdot I_2 \cdot I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Since multiplying any identity matrix by itself always yields the identity matrix, we conclude:

I_n^k = I_n

for any integer

k

General Case for Any Size:

For an

n \ imes n

identity matrix:

I_n^2 = I_n \cdot I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix} = I_n

Extending this further, for any power

k

I_n^k = I_n

This confirms that the identity matrix remains unchanged under exponentiation.

Relation to Other Special Matrices

In linear algebra, we divide square matrices into types based on their structural properties.Oftentimes, these types overlap, and a single matrix may belong to more than one category simultaneously.
The structural properties that define these categories - such as symmetry, orthogonality, or triangularity - can coexist within the same matrix, making the classification system interconnected rather than mutually exclusive.

For a better overview of the relationships between different types of square matrices, visit this page.

Being square matrix with elements of the main diagonal equal to 1 and all the rest equal to 0, identity matrix also qualifies for:

Diagonal Matrix: The identity matrix has elements aᵢⱼ = 0 for all i ≠ j and aᵢᵢ = 1, which fits the definition of diagonal matrix requiring all non-diagonal elements to be zero.
This diagram shows the exact nature of relations between diagonal and identity matrices (and other types ) visually.

Scalar Matrix: The identity matrix has equal elements λ = 1 on its main diagonal and zeros elsewhere, matching the form λI of scalar matrices.

Symmetric Matrix: For identity matrix, aᵢⱼ = aⱼᵢ for all i,j since all off-diagonal elements are 0 and diagonal elements are all 1.

Upper Triangular: The identity matrix has all elements below main diagonal equal to 0, satisfying the definition of upper triangular.

Lower Triangular: The identity matrix has all elements above main diagonal equal to 0, satisfying the definition of lower triangular.

On the other hand, Identity Matrix never satisfies definition of :

Zero Matrix: The zero matrix requires all elements to be 0, while identity matrix must have 1s on main diagonal.

Skew-Symmetric Matrix: A skew-symmetric matrix requires aᵢⱼ = -aⱼᵢ for all i,j, which identity matrix violates with 1s on diagonal.

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Identity Matrix

Introduction

Definition and Notation

Basic Properties

Closure Properties

Multiplicative Role in Matrix Algebra

Powers of the Identity Matrix

Relation to Other Special Matrices