Problem 13 - 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 \(e^{A t}\). $$A=\left[\begin{array}{rr} 1 & 1 \\ -1 & -1 \end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix \(A\) is nilpotent with the smallest positive integer \(k=2\), as \(A^2\) is the zero matrix. Using Definition 7.4.1 and the fact that \(A\) is nilpotent, we can compute the exponential of \(A\) times \(t\) as \(e^{A t} = I + A t\), which results in \(e^{A t} = \left[\begin{array}{rr} 1+t & t \\\ -t & 1-t \end{array}\right]\).

Step 1: Determine if the Matrix is Nilpotent

To find the smallest positive integer k such that \(A^k=0\), we can multiply the matrix \(A\) by itself multiple times until we reach the zero matrix or observe a pattern. Start by computing \(A^2\): $$ A^2 = AA = \left[\begin{array}{rr} 1 & 1 \\\ -1 & -1 \end{array}\right] \left[\begin{array}{rr} 1 & 1 \\\ -1 & -1 \end{array}\right] = \left[\begin{array}{rr} 0 & 0 \\\ 0 & 0 \end{array}\right] $$ Since \(A^2\) is the zero matrix, we can conclude that the given matrix \(A\) is nilpotent and \(k=2\).

Step 2: Compute the Exponential of \(A\) times \(t\) using Definition 7.4 .1

Definition 7.4.1 states that the exponential of a matrix is defined as: $$e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots$$ However, since \(A\) is a nilpotent matrix with \(A^2 = 0\), all subsequent matrix powers of \(A\) greater than or equal to 2 will also be 0. This means that the exponential function becomes: $$e^{At} = I + At$$ Now, we just need to calculate the matrix \(At\) to find \(e^{At}\).

Step 3: Calculate \(At\)

To find \(At\), simply multiply the matrix \(A\) by the scalar \(t\): $$ At = t\left[\begin{array}{rr} 1 & 1 \\\ -1 & -1 \end{array}\right]= \left[\begin{array}{rr} t & t \\\ -t & -t \end{array}\right] $$

Step 4: Compute \(e^{At}\)

Now, we can calculate \(e^{At}\): $$e^{At} = I + At = \left[\begin{array}{rr} 1 & 0 \\\ 0 & 1 \end{array}\right]+ \left[\begin{array}{rr} t & t \\\ -t & -t \end{array}\right]= \left[\begin{array}{rr} 1+t & t \\\ -t & 1-t \end{array}\right] $$ Thus, the exponential of the given matrix \(A\) times \(t\) is: $$e^{At}=\left[\begin{array}{rr} 1+t & t \\\ -t & 1-t \end{array}\right]$$

Key Concepts

Matrix Exponential

The matrix exponential is a concept that extends the idea of the exponential function from numbers to matrices. When dealing with matrices, especially in the realm of solving differential equations or quantum mechanics, the matrix exponential plays a pivotal role. It allows us to define expressions of the form \( e^{At} \), where \( A \) is a matrix and \( t \) is a scalar.

For matrix \( A \), the exponential \( e^{At} \) is computed as:

• \( e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots \)

In this series, \( I \) is the identity matrix, and the terms \((At)^n/n!\) involve higher powers of the matrix \( A \), but with factorials in the denominator to manage convergence, much like the classic series for \( e^x \).

For nilpotent matrices, like our given matrix \( A \), higher powers eventually become zero. Since \( A^2 = 0 \) for our matrix, the series simplifies dramatically:

• \( e^{At} = I + At \)

So, calculating the matrix exponential for nilpotent matrices often involves only a few simple computations.

Matrix Powers

Matrix powers involve repeatedly multiplying a matrix by itself. An important aspect of working with matrix powers is understanding how they affect different types of matrices, such as nilpotent matrices.

A nilpotent matrix is one that, when raised to some power, results in the zero matrix. In mathematical terms, a matrix \( A \) is nilpotent if there exists a positive integer \( k \) such that \( A^k = 0 \).

• For our given matrix \( A \), when we calculated \( A^2 \), we found that it equals the zero matrix.

Hence, \( k = 2 \) is the smallest such integer for this matrix. This behavior is critical because, for exponentials of matrices, it implies simplification in exponential series due to vanishing higher powers.

It's interesting to note that not all matrices exhibit this behavior; in fact, most don't, which makes nilpotent matrices quite special in linear algebra.

2x2 Matrix

A 2x2 matrix is a simple yet powerful representation in linear algebra. It is composed of two rows and two columns, resulting in four elements. The matrix we are analyzing, for example, is:

• \( \begin{bmatrix} 1 & 1 \ -1 & -1 \end{bmatrix} \)

These matrices can represent linear transformations in two-dimensional space, making them useful for various applications such as in physics and computer graphics.

In our example, this 2x2 matrix is also a nilpotent matrix, illustrating how it can be both compact and carry interesting properties. By calculating powers or exponentials of such matrices, we explore deeper algebraic structures.

Working with 2x2 matrices often provides insights into larger, more complex systems, while still allowing relatively straightforward computations. They're perfect entry points for students new to the subject.