Question

The given matrix \(A\) is diagonalizable. (a) Find \(T\) and \(D\) such that \(T^{-1} A T=D\). (b) Using (12c), determine the exponential matrix \(e^{A t}\).\(A=\left[\begin{array}{rr}0 & 2 \\ -2 & 0\end{array}\right]\)

Short answer

Question: Determine the exponential matrix \(e^{At}\) for the matrix A, where $$ A = \left[\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right] $$ Solution: The exponential matrix \(e^{At}\) for the given matrix A is: $$ e^{At} = \left[\begin{array}{rr} e^{2t}-e^{-2t} & 2(e^{-2t}-e^{2t}) \\ 2(e^{2t}-e^{-2t}) & e^{2t}-e^{-2t} \end{array}\right] $$

Step-by-step solution

Find the eigenvalues and eigenvectors of matrix A

First, let's find the eigenvalues of matrix A. To do this, we will solve the characteristic equation \(|A-\lambda I|=0\), where \(I\) is the identity matrix, and \(\lambda\) is the eigenvalue. $$ \begin{vmatrix} 0-\lambda & 2 \\\ -2 & 0-\lambda \end{vmatrix}=(0-\lambda)(0-\lambda)-(-2)(2)=\lambda^2-4 = 0 $$ Solve this equation for \(\lambda\), we get \(\lambda = \pm 2\). Thus, the eigenvalues are 2 and -2. Now, let's find the corresponding eigenvectors: For \(\lambda = 2\), the equation to solve is: $$ (A-\lambda I)v= \left[\begin{array}{rr}-2 & 2 \\\ -2 & -2\end{array}\right] \begin{bmatrix}v_1 \\ v_2\end{bmatrix}=\begin{bmatrix}0 \\ 0\end{bmatrix} $$ We can see from the equation above that the eigenvector is v = (-1,1). For \(\lambda = -2\), the equation to solve is: $$ (A-\lambda I)v=\left[\begin{array}{rr}2 & 2 \\\ -2 & 2\end{array}\right] \begin{bmatrix}v_1 \\ v_2\end{bmatrix}=\begin{bmatrix}0 \\ 0\end{bmatrix} $$ We can see from the equation above that the eigenvector is v = (1,1).

Formulate diagonal matrix D and matrix T

Now, let's construct matrix D and matrix T. The diagonal matrix D is formed by placing the eigenvalues of matrix A along its diagonal, and the matrix T is formed by placing the corresponding eigenvectors in its columns. So, we have: $$ D = \left[\begin{array}{rr} 2 & 0 \\ 0 & -2 \end{array}\right]\text{, and } T = \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] $$

Calculate the inverse of matrix T

To compute the inverse of matrix T, you can use the formula for a 2x2 matrix: $$ T^{-1} = \frac{1}{\text{det}(T)}\left[\begin{array}{rr} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array}\right] $$ Calculate the determinant of T: $$ \text{det}(T)=-1(1)-(1)(1)=-2 $$ Therefore, we have: $$ T^{-1} = \frac{1}{-2}\left[\begin{array}{rr} 1 & -1 \\ -1 & -1 \end{array}\right]=\left[\begin{array}{rr} -0.5 & 0.5 \\ 0.5 & 0.5 \end{array}\right] $$

Verify the relationship \(T^{-1}AT = D\)

Now, let's verify the relationship \(T^{-1}AT = D\). We need to perform the matrix multiplications and check if it equals D. $$ T^{-1}AT = \left[\begin{array}{rr} -0.5 & 0.5 \\ 0.5 & 0.5 \end{array}\right] \left[\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right] \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right] = \left[\begin{array}{rr} 2 & 0 \\ 0 & -2 \end{array}\right] $$ As you can see, \(T^{-1}AT = D\), which confirms that the matrices we found are correct.

Find the exponential matrix \(e^{At}\)

To calculate the exponential matrix \(e^{At}\), we can use the formula (12c): \(e^{At} = T e^{Dt} T^{-1}\). First, let's calculate the matrix \(e^{Dt}\): $$ e^{Dt} = \left[\begin{array}{rr} e^{2t} & 0 \\ 0 & e^{-2t} \end{array}\right] $$ Now, let's substitute this into the formula and calculate the exponential matrix \(e^{At}\): $$ e^{At} = T e^{Dt} T^{-1} = \left[\begin{array}{rr} -1 & 1 \\ 1 & 1 \end{array}\right]\left[\begin{array}{rr} e^{2t} & 0 \\ 0 & e^{-2t} \end{array}\right]\left[\begin{array}{rr} -0.5 & 0.5 \\ 0.5 & 0.5 \end{array}\right]=\left[\begin{array}{rr} e^{2t}-e^{-2t} & 2(e^{-2t}-e^{2t}) \\ 2(e^{2t}-e^{-2t}) & e^{2t}-e^{-2t} \end{array}\right] $$ So, the exponential matrix \(e^{At}\) is: $$ e^{At} = \left[\begin{array}{rr} e^{2t}-e^{-2t} & 2(e^{-2t}-e^{2t}) \\ 2(e^{2t}-e^{-2t}) & e^{2t}-e^{-2t} \end{array}\right] $$

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is crucial for various matrix operations and applications in linear algebra. These concepts are foundational in finding the matrix diagonalization process. Let's dive into what these terms mean.

An eigenvalue is a special scalar that, when multiplied by a specific vector, does not change the direction of that vector, only its magnitude. This special vector is known as an eigenvector. To find eigenvalues, we solve the characteristic equation, derived from the matrix in question. It typically has the form \( \text{det}(A - \lambda I) = 0 \), where \( A \) is our matrix, \( I \) is the identity matrix, and \( \lambda \) represents the eigenvalues.

Upon finding the eigenvalues, we can find the corresponding eigenvectors by solving \( (A - \lambda I)\vec{v} = \vec{0} \), where \( \vec{v} \) is the eigenvector. Eigenvectors associated with different eigenvalues are linearly independent, and these pairs reveal important properties about the transformation represented by the matrix, such as its principal directions of stretching or compressing.

Exponential Matrix

The exponential matrix, denoted as \( e^{At} \), is another fascinating concept in linear algebra, typically used for solving systems of linear differential equations. Calculating the matrix exponential involves taking a matrix, \( A \), to a power driven by a scalar, \( t \), and dealing with infinite series. However, the process simplifies significantly when \( A \) is diagonalizable.

For a diagonalizable matrix, we first find a diagonal matrix \( D \) and a transformation matrix \( T \) such that \( T^{-1}AT = D \). With these in hand, the exponential matrix is given by the formula \( e^{At} = T e^{Dt} T^{-1} \), where \( e^{Dt} \) is easy to compute since \( D \) is diagonal; we simply exponentiate the diagonal elements. This process transforms complex operations into simpler forms, showcasing the power of matrix diagonalization.

Matrix Diagonalization Process

Matrix diagonalization is a transformative process that simplifies matrix operations. It involves finding a diagonal matrix \( D \) that is equivalent to the original matrix \( A \) within a change of basis. The process relies on our ability to find the eigenvalues and eigenvectors of \( A \).

A diagonalizable matrix can be expressed as \( A = TDT^{-1} \), where \( T \) is the matrix of eigenvectors and \( D \) contains eigenvalues on its diagonal. The advantage of diagonalization is that it turns complex matrix functions, such as raising a matrix to a power or finding matrix exponentials, into much simpler tasks, since these operations are straightforward on a diagonal matrix. This is crucial when dealing with large or complicated systems in computational applications.

Matrix Operations

Matrix operations include a variety of mathematical procedures that can be performed on matrices, including addition, subtraction, multiplication, and inversion. In the context of diagonalization, the matrix multiplication and inversion play key roles.

• Matrix Multiplication: This is a way to combine two matrices to produce a third matrix. However, it is not commutative, meaning that \( AB \) does not necessarily equal \( BA \).
• Matrix Inversion: The process of finding a matrix \( B \) such that when it is multiplied with the original matrix \( A \) yields the identity matrix \( I \) is known as finding the inverse, denoted as \( A^{-1} \). However, not all matrices are invertible.

These operations are essential in the process of diagonalization and must be understood for accurate computation and application of various linear transformations and solving systems of equations.