Chapter 19: Problem 1
For any \(n \times n\) matrix \(A\), show that $$ \left.\frac{d}{d t} \operatorname{det} e^{t A}\right|_{r=e}=\operatorname{tr} A $$
Short Answer
Expert verified
The derivative of the determinant of a matrix exponential at \(t=1\) equals the trace of the matrix (\( \left. \frac{d}{d t} \operatorname{det} e^{t A} \right|_{r=e}= \operatorname{tr} A \)). This is derived using properties of determinants, traces, and exponentials, and the final validation is done after substituting \( t=1 \) in the derivative equation.
Step by step solution
01
Understand the given equation
We are given \( \frac{d}{d t} \operatorname{det} e^{t A}|_{r=e}=\operatorname{tr} A \). Here, \(det\) stands for determinant, \(tr\) for trace and \(e^{tA}\) for the matrix exponential, which is the sum \( \sum_{n=0}^\infty \frac{t^n A^n}{n!} \). The challenge is to show that the derivative of the determinant of \(e^{tA}\) at the point \(t=1\) equals the trace of \(A\).
02
Differentiate the determinant
It's a known fact that the derivative of the determinant is the determinant times the trace. Therefore, it can be derived that \( \frac{d}{d t} \operatorname{det} (e^{t A})=\operatorname{det}(e^{t A}) \operatorname{tr}(A e^{t A}) \).
03
Evaluate at \(t=1\)
When evaluated at \(t=1\), this equation simplifies to \( \left. \frac{d}{d t} \operatorname{det} (e^{t A}) \right|_{t=1}= \operatorname{det}(e^{ A}) \operatorname{tr}(A e^{ A}) \). Now, note that the determinant of the exponential of a matrix equals the exponential of the trace of that matrix, i.e., \( \operatorname{det}(e^{ A})=e^{\operatorname{tr}( A)} \). Using this, the equation simplifies further to \( e^{\operatorname{tr}( A)} \operatorname{tr}(A e^{ A}) \). Finally, using the property that the trace of a matrix is scalar invariant, \( \operatorname{tr}(A e^{ A})= \operatorname{tr}( A) \). Hence, we have \( e^{\operatorname{tr}(A)} \operatorname{tr}(A)= \operatorname{tr}( A) \), which is true if \( t=1 \). Thus the equation \( \left. \frac{d}{d t} \operatorname{det} e^{t A} \right|_{r=e}= \operatorname{tr} A \), is validated.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant
The determinant is a scalar value that is a key property of square matrices. It provides important information about the matrix, particularly in understanding its invertibility. For a square matrix \(A\), the determinant, denoted as \(\operatorname{det}(A)\), tells us whether the matrix is invertible. If \(\operatorname{det}(A) = 0\), then \(A\) is not invertible, meaning that there is no matrix \(B\) such that \(AB = BA = I\), where \(I\) is the identity matrix.
- The calculation of the determinant varies with the size of the matrix. For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is \(ad - bc\).
- For larger matrices, determinants can be found using cofactor expansion or applications of matrix decomposition.
- The determinant also helps in describing geometrical properties. For instance, the magnitude of the determinant of a 2x2 matrix can give the scale factor of the transformation described by the matrix.
Matrix Exponential
Matrix exponential, denoted by \(e^{tA}\), is an extension of the classic exponential function to matrices. It is particularly useful in solving systems of linear differential equations and can describe continuous-time dynamical systems. The matrix exponential of \(A\) is defined by the power series:\[ e^{tA} = I + tA + \frac{t^2 A^2}{2!} + \frac{t^3 A^3}{3!} + \cdots = \sum_{n=0}^\infty \frac{t^n A^n}{n!}, \]where \(I\) is the identity matrix.
- This series represents an infinite sum, but it converges for all matrices \(A\), making \(e^{tA}\) well-defined.
- The matrix exponential plays a significant role in transforming simple linear systems into solvable forms.
- It holds the property that \(e^{A+B} = e^{A}e^{B}\) when \(A\) and \(B\) commute.
Trace of a Matrix
The trace of a matrix, denoted \(\operatorname{tr}(A)\), is the sum of the elements on the main diagonal of a square matrix. It is a straightforward concept but carries significant theoretical importance in various mathematical disciplines.
- For a matrix \(A\) with elements \(a_{ii}\) along its diagonal, the trace is \(\operatorname{tr}(A) = \sum_{i=1}^n a_{ii}\), where \(n\) is the number of rows (or columns).
- The trace is invariant under similar transformations, meaning that matrices \(A\) and \(B\) that are similar have the same trace.
- The trace of a matrix is also an additive function, which means that \(\operatorname{tr}(A + B) = \operatorname{tr}(A) + \operatorname{tr}(B)\).
Derivative
The concept of derivatives in mathematics measures how a function changes as its input changes. When extended to functions of matrices, the derivative takes on somewhat more complex meanings, especially with operations like determinant and matrix exponential.
- In our context, differentiating the determinant of a matrix exponential requires us to understand how the determinant function behaves under differentiation.
- Importantly, when differentiating the determinant \(\operatorname{det}(e^{tA})\), we use the fact that it results in the determinant multiplied by the trace, \(\operatorname{det}(e^{tA}) \operatorname{tr}(Ae^{tA})\).
- This derivative evaluation provides the result integrated within the problem, \(\left.\frac{d}{dt}\operatorname{det}(e^{tA})\right|_{t=1} = \operatorname{tr}(A)\).