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 \(e^{A t}\).\(A=\left[\begin{array}{rr}-2 & 2 \\ 0 & 3\end{array}\right]\)

Short answer

Question: Determine the exponential matrix \(e^{At}\) for the given matrix \(A = \begin{bmatrix} -2 & 2 \\ 0 & 3 \end{bmatrix}\), where \(t\) is a real variable. Answer: The exponential matrix \(e^{At}\) can be calculated as \(e^{At} = \begin{bmatrix} 2e^{3t} & -\frac{1}{5}e^{-2t} + \frac{2}{5}e^{3t} \\ 5e^{3t} & \frac{1}{5}e^{3t} \end{bmatrix}\).

Step-by-step solution

Find the eigenvalues of matrix A

First, we need to find the eigenvalues of the given matrix \(A\). For this, we solve the characteristic equation: \(\text{det}(A - \lambda I) = 0\) \((-2-\lambda)(3-\lambda)-2\cdot 0=0\) Simplification gives: \(\lambda^2-\lambda-6=0\) The eigenvalues can be found by solving this quadratic equation, getting \(\lambda_1 = 3\) and \(\lambda_2 = -2\).

Find the eigenvectors of matrix A

Having found the eigenvalues, we now need to find the eigenvectors associated with them. (a) For \(\lambda_1 = 3\): Solve \((A - \lambda_1 I) \vec{v} = 0\) \(\begin{bmatrix} -5 & 2 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) From this, we find that \(-5x+2y=0\). The eigenvector for \(\lambda_1\) can be represented by \(\vec{v}_{1}=c_1\begin{bmatrix} 2 \\ 5 \end{bmatrix}\), where \(c_1\) is a constant. (b) For \(\lambda_2 = -2\): Solve \((A - \lambda_2 I) \vec{v} = 0\) \(\begin{bmatrix} 0 & 2 \\ 0 & 5 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) This yields the equation \(2y=0\), giving the eigenvector for \(\lambda_2\) as \(\vec{v}_{2}=c_2\begin{bmatrix} 1 \\ 0 \end{bmatrix}\), where \(c_2\) is a constant.

Construct T and D matrices

Now that we have the eigenvectors corresponding to the eigenvalues, we can form the matrices \(T\) and \(D\). \(T=\left[\begin{array}{cc}\vec{v}_{1} & \vec{v}_{2}\end{array}\right]=\) \(\begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix}\) The matrix \(D\) is a diagonal matrix with eigenvalues on the main diagonal: \(D=\begin{bmatrix} 3 & 0 \\ 0 & -2 \end{bmatrix}\)

Find the inverse of T

To find the inverse of matrix \(T\), we can use the formula: \(T^{-1} = \frac{1}{\text{det}(T)} \cdot \text{adj}(T)\) \(\text{det}(T) = (2)(0) - (1)(5) = -5\) \(T^{-1} = \frac{1}{-5} \cdot \begin{bmatrix} 0 & -1 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} 0 & \frac{1}{5} \\ 1 & -\frac{2}{5} \end{bmatrix}\)

Compute the exponential matrix \(e^{Dt}\)

Since matrix \(D\) is a diagonal matrix, finding the exponential matrix \(e^{Dt}\) is straightforward: \(e^{Dt} = \begin{bmatrix} e^{3t} & 0 \\ 0 & e^{-2t} \end{bmatrix}\)

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

To find the exponential matrix \(e^{At}\), we use the formula: \(e^{At} = T e^{Dt} T^{-1}\) \(= \begin{bmatrix} 2 & 1 \\ 5 & 0 \end{bmatrix} \begin{bmatrix} e^{3t} & 0 \\ 0 & e^{-2t} \end{bmatrix} \begin{bmatrix} 0 & \frac{1}{5} \\ 1 & -\frac{2}{5} \end{bmatrix}\) \(=\begin{bmatrix} 2 e^{3t} &e^{-2t} \\5e^{3t} &0 \end{bmatrix} \begin{bmatrix} 0 & \frac{1}{5} \\ 1 & -\frac{2}{5} \end{bmatrix}\) \(=\begin{bmatrix} 2e^{3t} & -\frac{1}{5}e^{-2t} + \frac{2}{5}e^{3t} \\ 5e^{3t} & \frac{1}{5}e^{3t} \end{bmatrix}\) So the exponential matrix \(e^{At}\) is: \(e^{At} = \begin{bmatrix} 2e^{3t} & -\frac{1}{5}e^{-2t} + \frac{2}{5}e^{3t} \\ 5e^{3t} & \frac{1}{5}e^{3t} \end{bmatrix}\)

Key concepts

eigenvalues and eigenvectors

In linear algebra, eigenvalues and eigenvectors are fundamental concepts used in numerous areas including matrix diagonalization. Simply put, an eigenvalue is a special scalar associated with a matrix such that when the matrix is multiplied by an eigenvector, it results in the eigenvector being scaled by this scalar. Mathematically, for a square matrix \(A\), if there exists a vector \(\vec{v}\) and a scalar \(\lambda\) such that \(A\vec{v} = \lambda \vec{v}\), then \(\lambda\) is called an eigenvalue of \(A\) and \(\vec{v}\) is the corresponding eigenvector.

To find the eigenvalues of a matrix \(A\), we solve the characteristic equation \(\text{det}(A - \lambda I) = 0\), where \(I\) is the identity matrix. This process expresses the eigenvalue \(\lambda\) as the root of a polynomial. With the eigenvalues known, finding eigenvectors involves solving the equation \((A - \lambda I) \vec{v} = 0\), which is a homogeneous system of linear equations.

Understanding eigenvalues and eigenvectors is crucial for applications such as the diagonalization of matrices since the eigenvectors form the basis for the transformation of the matrix into its diagonal form. This is essential for simplifying matrix operations, such as those involved in solving differential equations or computing exponential matrices.

exponential matrix

An exponential matrix is a concept mostly used to solve systems of linear differential equations. The matrix exponential \(e^{At}\) is crucial in contexts like stability analysis of dynamic systems. It is obtained from a matrix \(A\) and serves a similar role to the scalar exponential function \(e^t\) but applied to matrices. The exponential matrix is defined by the infinite series:

\[ e^{At} = I + At + \frac{A^2 t^2}{2!} + \frac{A^3 t^3}{3!} + \cdots \]

However, when a matrix \(A\) is diagonalizable, computing \(e^{At}\) becomes more straightforward by utilizing the process of matrix diagonalization. We find matrices \(T\) and \(D\) such that \(T^{-1}AT = D\), a diagonal matrix. In this form, computing \(e^{Dt}\) is straightforward as it only involves exponentiating each of the eigenvalues found on the diagonal of \(D\).

Thus, the exponential matrix can be computed as \(e^{At} = Te^{Dt}T^{-1}\), which makes the task computationally simpler. This ability to simplify calculations makes exponential matrices a powerful tool in engineering and physics, especially in modeling the behavior of time-dependent systems.

diagonal matrix

A diagonal matrix is a type of matrix where all entries outside the main diagonal are zero. This simplicity allows many matrix operations to be performed more easily. If a matrix is diagonal, it implies that the transformation it represents does not involve any mixing of different coordinates or basis elements.

The importance of a diagonal matrix shines in the context of matrix diagonalization. When a matrix\( A \) is diagonalizable, it can be expressed in the form \( A = TDT^{-1}\), where \(D\) is a diagonal matrix comprised of eigenvalues from \(A\) and \(T\) is the matrix of corresponding eigenvectors. Operations on \(D\) are simpler, such as finding powers, because they involve manipulating just the diagonal elements.

In practice, this means if you have a diagonal matrix like \(D = \text{diag}(d_1, d_2, \ldots, d_n)\), then:\-\(D^k = \text{diag}(d_1^k, d_2^k, \ldots, d_n^k)\), where \(k\) is any power. -Calculating its inverse only requires inverting each diagonal element, if none are zero.

These properties make diagonal matrices exceedingly useful in simplifying algebraic computations. The diagonalization process also plays a crucial role in understanding concepts like the exponential matrix, where direct computation can be challenging without simplification.

inverse matrix

The inverse of a matrix, when it exists, is a key concept that allows us to "undo" the effects of a matrix multiplication. For a square matrix \(A\), its inverse is denoted as \(A^{-1}\) and is characterized by the property: \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix.

Finding an inverse is vital in many applications, notably in solving equations of the form \(Ax = b\), where \(x = A^{-1}b\) provides a solution if \(A^{-1}\) exists. A matrix is invertible only if it is non-singular, meaning its determinant is non-zero.

The process of finding the inverse of a matrix can often be direct if the matrix is simple, such as a 2x2 matrix, through a formula involving its determinant and cofactor matrix. For matrices of larger size, more sophisticated methods like row reduction or adjoint methods are necessary.
In the context of diagonalization, the matrix \(T^{-1}\) is essential. It facilitates transforming a matrix \(A\) into its diagonal form via \(T^{-1}AT\), from which exponential matrices can be easily computed. The ability to find and use inverse matrices is fundamental in various fields, including solving linear systems and performing change of basis in vector spaces.