Let A be a 2 ×2 matrix such that tr A=0 and δ=√(det A)>0. (a) Prove that e^A=(cosδ) I+(sinδ)/(δ)

step 1 Hello there. So let's suppose that we have a matrix that is a square matrix and we need to prove

Let A be a 2 ×2 matrix such that tr A=0 and δ=√(det A)>0....

Let $A$ be a $2 \times 2$ matrix such that $\operatorname{tr} A=0$ and $\delta=\sqrt{\operatorname{det} A}>0$. (a) Prove that $e^A=(\cos \delta) \mathrm{I}+\frac{\sin \delta}{\delta}$ A. Hint: Use Exercise 8.2.52. (b) Establish a similar formula when $\operatorname{det} A<0$. (c) What if $\operatorname{det} A=0$ ?

Let $A$ be a $2 \times 2$ matrix such that $\operatorname{tr} A=0$ and $\delta=\sqrt{\operatorname{det} A}>0$.
(a) Prove that $e^A=(\cos \delta) \mathrm{I}+\frac{\sin \delta}{\delta}$ A. Hint: Use Exercise 8.2.52.
(b) Establish a similar formula when $\operatorname{det} A<0$. (c) What if $\operatorname{det} A=0$ ?

Key Concepts

Matrix Exponential

The matrix exponential is a fundamental concept defined by a power series expansion analogous to the exponential function for real numbers. It provides a way to solve systems of linear differential equations, compute functions of matrices, and study dynamical systems. The matrix exponential, e^A, is defined as the infinite sum I + A + (1/2!)A^2 + (1/3!)A^3 + …, and its properties can be exploited especially when the matrix has special structures, such as having a zero trace or being nilpotent.

Power Series Expansion

The power series expansion of the exponential function is used to compute the exponential of a matrix by substituting the matrix into the series. When the matrix satisfies certain conditions (for example, when its powers can be expressed in terms of lower powers via the Cayley-Hamilton theorem), the series can be rearranged into familiar functions like cosine, sine, hyperbolic cosine, or hyperbolic sine, leading to closed-form formulas for e^A that greatly simplify computations.

Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial. This theorem is particularly useful for 2 × 2 matrices because it implies that higher powers of the matrix can be expressed as linear combinations of the identity matrix and the matrix itself. This reduction allows one to rewrite the matrix exponential (and other matrix functions) in terms of a finite number of terms, leading to simplified formulas under various conditions on the determinant and trace.

Trigonometric and Hyperbolic Functions in Matrix Functions

When evaluating the matrix exponential, the series expansion can sometimes be grouped to resemble the expansions of cosine and sine (or hyperbolic cosine and sine). This typically happens when the square of the matrix is proportional to the identity matrix. If the constant of proportionality is negative, the series naturally leads to trigonometric functions, while if it is positive, one ends up with hyperbolic functions. These representations connect the behavior of the matrix exponential to well?known oscillatory or exponential growth/decay properties.

Nilpotent Matrices and Special Cases

Special cases such as nilpotent matrices (where a certain power of the matrix is zero) often arise in the context of the matrix exponential. For example, if the determinant condition forces the matrix to be nilpotent, the power series for e^A terminates after a finite number of terms, yielding particularly simple and elegant formulas. This analysis is crucial when classifying different cases based on determinant and trace, as each scenario (positive, negative, or zero determinant) can lead to fundamentally different behavior of the exponential function.

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