Question

Show that if \(D\) is a diagonal matrix, then \(D^{n}\) is the diagonal matrix with elements equal to the \(n^{\text {th }}\) power of the elements of \(\mathrm{D}\). Also show that if \(\mathrm{D}=\mathrm{C}^{-1} \mathrm{MC}\), then \(\mathrm{D}^{n}=\mathrm{C}^{-1} \mathrm{M}^{n} \mathrm{C},\) so \(\mathrm{M}^{n}=\mathrm{CD}^{n} \mathrm{C}^{-1} .\) Hint: For \(n=2,\left(\mathrm{C}^{-1} \mathrm{MC}\right)^{2}=\mathrm{C}^{-1} \mathrm{MCC}^{-1} \mathrm{MC}\) what is \(\mathrm{CC}^{-1} ?\)

Short answer

If D is a diagonal matrix, then \( D^n \) has elements equal to the n-th power of D's elements. Also, \( D^n = C^{-1}M^nC \) and \( M^n = CD^nC^{-1} \).

Step-by-step solution

Understanding Diagonal Matrices

A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. If a diagonal matrix is denoted as D, then: \[ D = \begin{pmatrix} d_1 & 0 & 0 & \ 0 & d_2 & 0 & \ 0 & 0 & d_3 & \ & & & \ddots \end{pmatrix} \] Each element on the diagonal can be represented as \( d_i \).

Calculating Powers of a Diagonal Matrix

For a diagonal matrix D, the power \( D^n \) can be found by raising each of its diagonal elements to the n-th power individually: \[ D^n = \begin{pmatrix} d_1^n & 0 & 0 & \ 0 & d_2^n & 0 & \ 0 & 0 & d_3^n & \ & & & \ddots \end{pmatrix} \]

Given Matrix D as Conjugate to M

It is given that the diagonal matrix D can be represented as \( D = C^{-1}MC \), where C is an invertible matrix. To show \( D^n = C^{-1}M^nC \), we first check for \( n = 2 \):

Showing for n=2

For n=2, \( (C^{-1}MC)^2 \) can be written as: \[ (C^{-1}MC)^2 = (C^{-1}MC)(C^{-1}MC) = C^{-1}(MC)(C^{-1}MC) Since CC^{-1} = I \] the result simplifies to: \[ (C^{-1}MC)^2 = C^{-1}MMC = C^{-1}M^2C \]

General Proof for Any n

Using induction or repeated application, it can be generalized that: \[ D^n = (C^{-1}MC)^n = C^{-1}(MC)(C^{-1}MC) = C^{-1}MCC^{-1}M \ldots C^{-1}MC \( n \text{ times} \) \] By the property \( CC^{-1} = I \), each pair of C and \( C^{-1} \) reduces to I, ultimately resulting in: \[ D^n = C^{-1}M^nC \]

Showing \( M^n = CD^nC^{-1} \)

From the previous conclusion, rearrange the formula to express \( M^n \): \[ D^n = C^{-1}M^nC \Rightarrow M^n = CD^nC^{-1} \]

Key concepts

Matrix Powers

In the world of linear algebra, an important concept to understand is the power of a matrix. When we talk about matrix powers, it simply means multiplying a matrix by itself a certain number of times.
For instance, if we have a matrix A and we want to find its second power, denoted as \(A^2\), we would multiply A by itself: \(A^2 = A \times A\).
This concept extends to higher powers, such as \(A^3 = A \times A \times A\), and so on.
When dealing with diagonal matrices, calculating matrix powers becomes particularly straightforward. A diagonal matrix is a matrix where all the entries outside the main diagonal are zero. To find the power of a diagonal matrix, simply raise each of the diagonal elements to the corresponding power.
For example:

• For matrix \(D\), its nth power is given by \( D^n \), where each diagonal element \(d_i\) is raised to the power \(n\).
• Mathematically, this can be represented as: \[ D^n = \begin{pmatrix} d_1^n & 0 & 0 & \ 0 & d_2^n & 0 & \ 0 & 0 & d_3^n & \ \ \ \ & \ \ & \ddots \ \ \end{pmatrix} \]

Understanding this property of diagonal matrices greatly simplifies computations in linear algebra.

Conjugate Matrices

The concept of conjugate matrices is fundamental when dealing with transformations and similarity in linear algebra.
A matrix \(M\) is said to be conjugate to a matrix \(D\) (a diagonal matrix) if there exists an invertible matrix \(C\) such that \(D = C^{-1}MC\). This relationship shows that \(M\) and \(D\) are similar matrices and have the same eigenvalues.
If we want to show this with matrix powers, we start with the expression: \(D = C^{-1}MC\). Now, to find \(D^n\) for any positive integer \(n\), we use the property of matrix powers:

• For \(n = 2\), we see that \((C^{-1}MC)^2 = C^{-1}MC \times C^{-1}MC\).
• Utilizing the fact that \(CC^{-1} = I\) (where \(I\) is the identity matrix), the expression simplifies to \(C^{-1}M^2C\).
• Extending this property by repeated application or induction, we get \((C^{-1}MC)^n = C^{-1}M^nC\).

This shows that taking the power of the diagonal matrix \(D\) corresponds to taking the power of the original matrix \(M\) conjugated by \(C\).
This property is highly useful in simplifying complex matrix computations and understanding transformations.

Linear Algebra

Linear algebra forms the backbone of many mathematical disciplines and its concepts are widely applied in engineering, physics, computer science, and more.
It deals with vector spaces and linear mappings between these spaces, which includes operations involving matrices. Some fundamental concepts in linear algebra include:

• Vectors and Vector Spaces: A vector is an element of a vector space, and these spaces form the stage where linear mappings act.
• Matrices: Matrices represent linear transformations and are used for various operations like addition, multiplication, and finding determinants.
• Eigenvalues and Eigenvectors: These are important for understanding the properties of matrices, especially in the context of eigenvalue decomposition and diagonalization.
• Diagonalization: This process involves finding a diagonal matrix \(D\) that is similar to a given matrix \(M\), which simplifies the computation of powers and other functions of matrices.

By mastering these core aspects of linear algebra, you can solve complex problems more efficiently and gain deeper insights into the behavior of linear systems. Understanding the properties of diagonal and conjugate matrices, as well as the implications of matrix powers, provides a robust foundation for further studies and applications.