Problem 37 - Advanced Engineering Mathematics

Question

If \(\mathbf{A}\) is an \(n \times n\) diagonalizablematrix, then \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\), where \(\mathbf{D}\) is a diagonal matrix. Show that if \(m\) is a positive integer, then \(\mathbf{A}^{m}=\mathbf{P D}^{m} \mathbf{P}^{-1}\).

Step-by-Step Solution

Answer

For any positive integer \(m\), \(\mathbf{A}^m = \mathbf{P} \mathbf{D}^m \mathbf{P}^{-1}\).

Step 1: Understand the Definitions

The problem states that \(\mathbf{A}\) is a diagonalizable matrix which means there exists an invertible matrix \(\mathbf{P}\) such that \(\mathbf{P}^{-1} \mathbf{A} \mathbf{P} = \mathbf{D}\), where \(\mathbf{D}\) is a diagonal matrix. Our goal is to show that for any positive integer \(m\), \(\mathbf{A}^m = \mathbf{P} \mathbf{D}^m \mathbf{P}^{-1}\).

Step 2: Establish the Diagonal Form of A

Given \(\mathbf{A} = \mathbf{P} \mathbf{D} \mathbf{P}^{-1}\), we can represent any power of \(\mathbf{A}\). Let's calculate \(\mathbf{A}^2\) to understand the process:\[ \mathbf{A}^2 = (\mathbf{P} \mathbf{D} \mathbf{P}^{-1})(\mathbf{P} \mathbf{D} \mathbf{P}^{-1}) \]Using the associative property of matrix multiplication:\[ \mathbf{A}^2 = \mathbf{P} \mathbf{D} (\mathbf{P}^{-1} \mathbf{P}) \mathbf{D} \mathbf{P}^{-1} \]Since \(\mathbf{P}^{-1} \mathbf{P} = \mathbf{I}\) (Identity Matrix):\[ \mathbf{A}^2 = \mathbf{P} \mathbf{D}^2 \mathbf{P}^{-1} \]

Step 3: Generalize to Any Power m

Observe that through similar steps, \(\mathbf{A}^3 = \mathbf{P} \mathbf{D}^2 \mathbf{P}^{-1} \mathbf{P} \mathbf{D} \mathbf{P}^{-1} = \mathbf{P} \mathbf{D}^3 \mathbf{P}^{-1}\). By induction, the pattern extends to:\[ \mathbf{A}^m = \mathbf{P} \mathbf{D}^m \mathbf{P}^{-1} \]For any integer \(m\), this holds because the identity matrix and associative property assist in collapsing repeated \(\mathbf{P}^{-1}\mathbf{P}\) pairs.

Step 4: Conclude the Solution

Now, each power of \(\mathbf{A}\) simply requires one to raise the diagonal matrix \(\mathbf{D}\) to that power and use the transformation \(\mathbf{P} \cdots \mathbf{P}^{-1}\). Hence, it is proven that \(\mathbf{A}^m = \mathbf{P} \mathbf{D}^m \mathbf{P}^{-1}\) for any positive integer \(m\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves combining two matrices to produce another matrix. Suppose you have two matrices \(\mathbf{A}\) and \(\mathbf{B}\), with dimensions \(n \times m\) and \(m \times p\), respectively. To multiply them, each element of the resulting matrix is a dot product of a row from \(\mathbf{A}\) and a column from \(\mathbf{B}\). It's important to note that matrix multiplication:

• Is not commutative: \(\mathbf{A} \mathbf{B} eq \mathbf{B} \mathbf{A}\) in most cases.
• Associative: \((\mathbf{A} \mathbf{B}) \mathbf{C} = \mathbf{A} (\mathbf{B} \mathbf{C})\).
• Distributes over addition: \(\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A} \mathbf{B} + \mathbf{A} \mathbf{C}\).

Matrix multiplication is essential for operations involving transforms like the one observed in the exercise, where the diagonalization of a matrix relies on multiplying matrices \(\mathbf{P}^{-1}\), \(\mathbf{A}\), and \(\mathbf{P}\) to achieve a diagonal matrix \(\mathbf{D}\). Understanding how these products work lets you explore more advanced operations like powering a matrix. It also helps in multiple applications like solving systems of equations and transformations in geometry.

Diagonal Matrix

A diagonal matrix is a special type of matrix where the elements outside the main diagonal are all zeros. The main diagonal itself consists of potentially non-zero elements. Formally, if \(\mathbf{D}\) is a diagonal matrix, then \(d_{ij} = 0\) for all \(i eq j\). Diagonal matrices have unique properties that make computations particularly simple:

• Easy Matrix Powers: For any integer \(m\), \(\mathbf{D}^m\) is simply raising each diagonal element to the \(m^{th}\) power.
• Simplifies Matrix Multiplication: When multiplying a diagonal matrix with another matrix, only the elements corresponding to the diagonal entries are affected.
• Eigenvalues Directly Visible: The eigenvalues of a diagonal matrix are the diagonal elements themselves.

In the context of diagonalizable matrices, achieving a form where \(\mathbf{D}\) is diagonal from \(\mathbf{A} = \mathbf{PDP}^{-1}\) is extremely advantageous. It greatly simplifies calculations involving powers of \(\mathbf{A}\), such as finding \(\mathbf{A}^m\). This ease comes from the straightforward process of dealing with single operations restricted to the diagonal, while multiplications are less computationally demanding.

Invertible Matrix

An invertible matrix, or non-singular matrix, is one that has an inverse. This means that for a square matrix \(\mathbf{A}\), there exists another matrix \(\mathbf{A}^{-1}\) such that:

• \(\mathbf{AA}^{-1} = \mathbf{I}\)
• \(\mathbf{A}^{-1}\mathbf{A} = \mathbf{I}\)

Here, \(\mathbf{I}\) is the identity matrix, acting similarly to the number 1 in arithmetic multiplication. The presence of an inverse is vital for transformations and solving linear equations. When a matrix is invertible:

• Its determinant is non-zero.
• It allows reversing the transformation it represents.
• It forms the backbone of matrix decompositions, like in the diagonalization process described.

In the problem you're dealing with, the invertibility of \(\mathbf{P}\) is crucial for finding the diagonal form of \(\mathbf{A}\). By ensuring \(\mathbf{P}^{-1}\) exists, we can manipulate \(\mathbf{A}\) into an easier-to-handle form, as well as backtrack transformations by plugging back into \(\mathbf{PDP}^{-1}\). This makes invertible matrices a key concept in understanding linear transformations and their applications in solving complex problems.