Problem 38 - Advanced Engineering Mathematics

Question

The \(m\) th power of a diagonal matrix $$ \mathbf{D}=\left(\begin{array}{cccc} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & a_{n n} \end{array}\right) $$ is \(\mathbf{D}^{m}=\left(\begin{array}{cccc}a_{11}^{m} & 0 & \cdots & 0 \\ 0 & a_{22}^{m} & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & a_{n n}^{m}\end{array}\right)\) Use this result to compute $$ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right)^{4} $$

Step-by-Step Solution

Answer

The 4th power of the diagonal matrix is \( \left(\begin{array}{rrrr} 16 & 0 & 0 & 0 \\ 0 & 81 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 625 \end{array}\right) \).

Step 1: Identify the Diagonal Matrix

We are given the matrix \(\mathbf{D}\) which is a diagonal matrix:\[\left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \0 & 3 & 0 & 0 \0 & 0 & -1 & 0 \0 & 0 & 0 & 5 \end{array}\right)\]We need to compute \(\mathbf{D}^4\). Each diagonal element of this matrix is the base for our exponentiation.

Step 2: Apply the Power to Each Diagonal Element

According to the property of a diagonal matrix, to compute \(\mathbf{D}^m\), raise each diagonal element to the power of \(m\). Here, \(m = 4\), so apply this to each element:- Raise 2 to the power of 4: \[2^4 = 16\]- Raise 3 to the power of 4: \[3^4 = 81\]- Raise -1 to the power of 4: \[(-1)^4 = 1\]- Raise 5 to the power of 4: \[5^4 = 625\]

Step 3: Construct the Resulting Matrix

Now construct the new matrix \(\mathbf{D}^4\) using the calculated powers of each diagonal element.\[\mathbf{D}^4 = \left(\begin{array}{rrrr} 16 & 0 & 0 & 0 \0 & 81 & 0 & 0 \0 & 0 & 1 & 0 \0 & 0 & 0 & 625 \end{array}\right)\]This is the resulting matrix after computing the 4th power of the given diagonal matrix.

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with the manipulation of matrices. A matrix is essentially a rectangular array of numbers or functions.
Matrices are used to represent and solve systems of linear equations among other things.
Matrix algebra involves several operations:

• Addition and Subtraction: Matrices of the same dimensions can be added or subtracted by adding or subtracting their corresponding elements.
• Multiplication: The product of two matrices is calculated by taking the dot product of rows and columns. Note that matrix multiplication is not commutative; for matrices A and B, generally, \(AB eq BA\).
• Scalar Multiplication: Multiply every element of a matrix by a scalar (a constant number).
• Transpose: The transpose of a matrix is obtained by swapping rows with columns.

Understanding these operations helps in exploring more complex concepts like diagonal matrices and their properties.
Each operation has specific rules, and knowing them is crucial for solving advanced mathematical problems.

Diagonal Matrix Properties

A diagonal matrix is a special type of matrix where all non-diagonal elements are zero. The main diagonal stores the significant values. This simple format grants diagonal matrices unique properties, making them very useful in many calculations.
Here are some basic properties:

• Simplicity of Inversion: Diagonal matrices are easily invertible. The inverse (if it exists) is simply the reciprocal of each non-zero diagonal element, placed in a similar diagonal matrix.
• Commutativity in Multiplication: Unlike general matrices, two diagonal matrices can be multiplied in any order.
• Determinant: The determinant of a diagonal matrix is the product of all its diagonal entries.
• Exponentiation: Raising a diagonal matrix to a power involves raising each of its diagonal elements to that power.

These properties simplify many matrix operations.
Particularly, exponentiation of diagonal matrices is straightforward and helps in complex calculations, such as matrix compound interest in financial calculations.

Matrix Exponentiation

Matrix exponentiation involves raising a matrix to a power, similar to numbers. While it's complex for general matrices, when dealing with diagonal matrices, the process is much simpler.
For a diagonal matrix, \[ \mathbf{D} = \begin{pmatrix} a_{11} & 0 & \cdots & 0 \ 0 & a_{22} & \cdots & 0 \ \vdots & & & \vdots \ 0 & 0 & \cdots & a_{nn} \end{pmatrix} \]raising it to a power \(m\) results in another diagonal matrix:\[ \mathbf{D}^m = \begin{pmatrix} a_{11}^m & 0 & \cdots & 0 \ 0 & a_{22}^m & \cdots & 0 \ \vdots & & & \vdots \ 0 & 0 & \cdots & a_{nn}^m \end{pmatrix} \]Each diagonal element is simply raised to the power, which dramatically simplifies the computation.
This property has practical applications:

• Dynamic Systems: Calculating future states of systems described by diagonal matrices.
• Finance: Computing compound interest with matrices.
• Physics: Modeling and predicting outcomes based on basic, state-aware movements.

In these contexts, knowing how to perform matrix exponentiation efficiently can save time and effort.