Lesson plan
Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by , , or , is the matrix given by the power series e^X = \sum_{k=0}^\infty \frac{1}{k!} X^k where X^0 is defined to be the identity matrix I with the same dimensions as X , and for integer . The series always converges, so the exponential of is well-defined. Equivalently, e^X = \lim_{k \rightarrow \infty} \left(I + \frac{X}{k} \right)^k for integer-valued , where is the identity matrix. Equivalently, the matrix exponential is provided by the solution Y(t)=e^{X t} of the (matrix) differential equation \frac d {dt} Y(t) = X\, Y(t), \quad Y(0) = I, evaluated at . When is an diagonal matrix then will be an diagonal matrix with each diagonal element equal to the ordinary exponential function applied to the corresponding diagonal element of .
Article summary
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix. The exponential of , denoted by , , or , is the matrix given by the power series e^X = \sum_{k=0}^\infty \frac{1}{k!} X^k where X^0 is defined to be the identity matrix I with the same dimensions as X , and for integer . The series always converges, so the exponential of is well-defined. Equivalently, e^X = \lim_{k \rightarrow \infty} \left(I + \frac{X}{k} \right)^k for integer-valued , where is the identity matrix. Equivalently, the matrix exponential is provided by the solution Y(t)=e^{X t} of the (matrix) differential equation \frac d {dt} Y(t) = X\, Y(t), \quad Y(0) = I, evaluated at . When is an diagonal matrix then will be an diagonal matrix with each diagonal element equal to the ordinary exponential function applied to the corresponding diagonal element of .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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