Problem 14 - Differential Equations and Linear Algebra

Question

An \(n \times n\) matrix \(A\) that satisfies \(A^{k}=0\) for some \(k\) is called nilpotent. Show that the given matrix is nilpotent, and use Definition 7.4 .1 to determine \(e^{A t}\). $$A=\left[\begin{array}{ll} -3 & 9 \\ -1 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix A is nilpotent with k = 2, as A^2 = 0. Using Definition 7.4.1, we can compute the matrix exponential e^(At). Since A^2 = 0, the formula for e^(At) simplifies to e^(At) = I + At. We find that: $$e^{At} =\left[\begin{array}{ll} 1 - 3t & 9t \\\ -t & 1 + 3t \end{array}\right]$$

Step 1: Determine if the given matrix A is nilpotent

First, we will multiply A by itself to find A^2, A^3, and so on and stop when we get a zero matrix or determine that A is not nilpotent. Given matrix A: $$A=\left[\begin{array}{ll} -3 & 9 \\\ -1 & 3 \end{array}\right]$$ Computing A^2: $$A^2 = AA = \left[\begin{array}{ll} -3 & 9 \\\ -1 & 3 \end{array}\right]\left[\begin{array}{ll} -3 & 9 \\\ -1 & 3 \end{array}\right] = \left[\begin{array}{ll} 0 & 0 \\\ 0 & 0 \end{array}\right]$$ Since A^2 is the zero matrix, we can say A is nilpotent and k = 2.

Step 2: Compute e^(At) using Definition 7.4.1

Now that we know A is a nilpotent matrix of order k=2, we will compute the matrix exponential of At using Definition 7.4.1. The formula for e^(At) is: $$e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + ...$$ Since we know A^2 = 0, all terms with A^2 and higher powers will be 0, hence can be discarded. Now, we have: $$e^{At} = I + At$$ where I is the identity matrix. Calculating At and e^(At): $$At = \left[\begin{array}{ll} -3 & 9 \\\ -1 & 3 \end{array}\right]t = \left[\begin{array}{ll} -3t & 9t \\\ -t & 3t \end{array}\right]$$ Now let's add the identity matrix I to At: $$e^{At} = I + At = \left[\begin{array}{ll} 1 & 0 \\\ 0 & 1 \end{array}\right] + \left[\begin{array}{ll} -3t & 9t \\\ -t & 3t \end{array}\right] = \left[\begin{array}{ll} 1 - 3t & 9t \\\ -t & 1 + 3t \end{array}\right]$$ Thus, the matrix exponential e^(At) is: $$e^{At} =\left[\begin{array}{ll} 1 - 3t & 9t \\\ -t & 1 + 3t \end{array}\right]$$

Key Concepts

Matrix Exponential

The matrix exponential is a fascinating concept in linear algebra that extends the idea of exponentiation from numbers to matrices. In particular, when dealing with a nilpotent matrix like the one given in the exercise, the matrix exponential simplifies beautifully. Normally, calculating the exponential of a matrix, denoted as e^(At), involves an infinite series. However, if matrix A is nilpotent, the series terminates early due to the higher powers of the matrix turning into the zero matrix.

For a typical matrix A, we would calculate e^(At) using the series:
e^(At) = I + At + (At)^2/2! + (At)^3/3! + ....
In our case, since A^2 = 0, all terms involving A^2 and higher powers vanish, leaving us with just e^(At) = I + At.

The identity matrix I acts as the starting point, and multiplying A by t effectively scales the original matrix by the scalar t. This results in a final matrix exponential of e^(At) that reflects only the linear growth over time, distinctly shown in the exercise solution.

Matrix Powers

Matrix powers are another cornerstone of linear algebra, especially when it comes to understanding nilpotent matrices. When we raise a matrix to a power, we are essentially performing repeated matrix multiplication. For a matrix A, A^2 would mean A multiplied by itself, A^3 would be A multiplied by A^2, and so on.

This concept is critical when identifying nilpotent matrices, as we seek the smallest integer k such that A^k equals the zero matrix. In the provided exercise, calculating A^2 immediately gave us the zero matrix, which confirmed A is nilpotent and we didn't need to calculate any higher powers. This highlights how some matrices, through their structure and properties, can simplify otherwise complex operations.

Linear Algebra

Linear algebra is a vast field that deals with vectors, matrices, and linear transformations. It’s essential in virtually every area of mathematics and applied sciences. The concepts of nilpotent matrices and matrix exponentials, as demonstrated in the exercise, are parts of linear algebra's fabric. Understanding these concepts helps in solving more complex problems in diverse areas like engineering, physics, computer science, and economics.

Nilpotent matrices are special because their powers eventually lead to a zero matrix, and this has implications for the behavior of systems described by such matrices. In practical terms, a nilpotent matrix might represent a system that dissipates energy over time until it stops working, or a process that fades away. The ease with which nilpotent matrices get along with operations like exponentiation illustrates the beauty of linear algebra: intricate-looking problems can sometimes have surprisingly straightforward solutions.