Scalar Matrix

Introduction

A scalar matrix is a special type of diagonal matrix where all elements on main diagonal are the same. This makes it a neat subset of diagonal matrices, inheriting their simplicity while adding an extra layer of uniformity. Since every scalar matrix is essentially a scaled version of the identity matrix, it plays a key role in scaling transformations and eigenvalue problems. With straightforward computations for multiplication, inversion, and determinants, scalar matrices are fundamental in linear algebra and mathematical applications.

Definition and Examples

A scalar matrix is a square matrix where all diagonal elements are equal to the same number and all elements outside the main diagonal are zeros.
Defined in this way, it falls under definition of diagonal matrix forming special case of this type.
Moreover, a scalar matrix can be seen as a scaled version of the identity matrix:identity matrix multiplied by a any number results in scalar matrix.

Mathematically, a scalar matrix

S

is defined as

(S)_{ij} = \begin{cases} c, & \text{if } i = j \\ 0, & \text{if } i \neq j \end{cases}

, where

c

is a constant scalar value.

Key Insight: Since a scalar matrix is just the identity matrix multiplied by a scalar, we can express it as

S = cI

, where

I

is the identity matrix and

c

is a scalar.

For an

n \times n

scalar matrix, we often use notation

S_{n\times n}

(or just

S

) where

n

is the order of the matrix.

2 \times 2

scalar matrix:

S_{2\times 2} = \begin{pmatrix} c & 0 \\ 0 & c \end{pmatrix}

3 \times 3

scalar matrix:

S_{3\times 3} = \begin{pmatrix} c & 0 & 0 \\ 0 & c & 0 \\ 0 & 0 & c \end{pmatrix}

General form for

n \times n

case:

S_{n\times n} = cI = \begin{pmatrix} c & 0 & 0 & \cdots & 0 \\ 0 & c & 0 & \cdots & 0 \\ 0 & 0 & c & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & c \end{pmatrix}

Basic Properties

A scalar matrix is a diagonal matrix where all diagonal elements are equal. This special form inherits key properties from diagonal matrices while adding additional features:

1. Structure: A scalar matrix by it's definition has identical values on its main diagonal, while all off-diagonal elements are zero.It is worth noting that the elements of the main diagonal may be equal to zeros as well and that kind of matrix, while being zero matrix, will be still refered as scalar diagonal type.

For further reference concerning different square matrices types and their relations you may visit this page.

Moreover, check this diagram illustrating various subsets of diagonal matrices and their intersections.

2. Multiplication: The product of two scalar matrices is another scalar matrix, where the diagonal value is the product of the original scalars.

See more detailed explanation about closure properties of scalar matrix here in dedicated section.

3. Commutativity: Scalar matrices commute with all diagonal matrices and with each other, meaning if

S_1

and

S_2

are scalar matrices, then

S_1 S_2 = S_2 S_1

4. Determinant: The determinant of an

n \times n

scalar matrix

S = cI

is given by

\det(S) = c^n

5. Trace: The trace of an

n \times n

scalar matrix is

\text{tr}(S) = nc

since all diagonal elements are

c

6. Inverse: A scalar matrix is invertible if

c \neq 0

. Its inverse is also scalar, given by

S^{-1} = (1/c)I

7. Eigenvalues and Eigenvectors: The only eigenvalue of a scalar matrix is

c

, with every nonzero vector as an eigenvector.

Closure Properties

A diagonal matrix exhibits specific closure properties under various operations:

1. Closed under Multiplication: The product of two diagonal matrices of the same size is another diagonal matrix: if

D_1

and

D_2

are diagonal, then

D_1 D_2

remains diagonal.

2. Closed under Exponentiation: Raising a diagonal matrix to any integer power preserves its diagonal structure: if

D

is diagonal, then

D^k

is also diagonal for any integer

k

.
This is pretty intuitive as exponentiation is a special case of multiplication. Check this section for more details.

3. Closed under Addition: The sum of two diagonal matrices is also diagonal, where each diagonal entry is the sum of corresponding entries: if

D_1

and

D_2

are diagonal, then

D_1 + D_2

is diagonal as well.

4. Closed under Scalar Multiplication: Multiplying a diagonal matrix by a scalar preserves its diagonal structure: if

D

is diagonal, then

cD

is also diagonal for any scalar

c

Powers of the Identity Matrix

Exponentiating a diagonal matrix is straightforward since diagonal matrices retain their structure under multiplication.

1. Power of a Diagonal Matrix

For any diagonal matrix

D = \text{diag}(d_{11}, d_{22}, \dots, d_{nn})

, raising it to the power of

k

results in:

D^k = \text{diag}(d_{11}^k, d_{22}^k, \dots, d_{nn}^k)

This follows from the fact that multiplying diagonal matrices is equivalent to exponentiating each diagonal element individually.

2. Comparison with Identity Matrix

The identity matrix is a special case of a diagonal matrix where all diagonal elements are 1:

I_n = \text{diag}(1,1,\dots,1)

Applying the same exponentiation rule:

I_n^k = \text{diag}(1^k, 1^k, \dots, 1^k) = I_n

Since 1 raised to any power is still 1, the identity matrix remains unchanged under exponentiation.

3. Example Calculation

Consider the diagonal matrix:

D = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}

Computing its square:

D^2 = \begin{pmatrix} 2^2 & 0 \\ 0 & 3^2 \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ 0 & 9 \end{pmatrix}

For a higher power, say

k = 3

D^3 = \begin{pmatrix} 2^3 & 0 \\ 0 & 3^3 \end{pmatrix} = \begin{pmatrix} 8 & 0 \\ 0 & 27 \end{pmatrix}

This confirms that exponentiation of diagonal matrices applies element-wise to the diagonal entries.

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