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- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/06%3A_Spectral_Theory/6.03%3A_Applications_of_Spectral_TheorySuppose we have a matrix A and we want to find A50 . One could try to multiply A with itself 50 times, but this is computationally extremely intensive (try it!). However diagonalization allows us...Suppose we have a matrix A and we want to find A50 . One could try to multiply A with itself 50 times, but this is computationally extremely intensive (try it!). However diagonalization allows us to compute high powers of a matrix relatively easily.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.04%3A_Applications_of_Spectral_Theory/6.4E%3A_Exercises_for_Section_6.4This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solvin...This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solving first-order systems through matrix exponentiation. Each exercise outlines problem setups, provides hints, and sometimes offers detailed solution computations.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/08%3A_Spectral_Theory/8.03%3A_Applications_of_Spectral_TheorySuppose we have a matrix A and we want to find A50 . One could try to multiply A with itself 50 times, but this is computationally extremely intensive (try it!). However diagonalization allows us...Suppose we have a matrix A and we want to find A50 . One could try to multiply A with itself 50 times, but this is computationally extremely intensive (try it!). However diagonalization allows us to compute high powers of a matrix relatively easily.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/05%3A_Matrix_Methods_for_Dynamical_Systems/5.04%3A_The_Backward-Euler_MethodThe utility of Equation is that it gives a means of solving for \tilde{x} at the present time, t, from the knowledge of \tilde{x} in the immediate past, t-dt. Of course we should expec...The utility of Equation is that it gives a means of solving for \tilde{x} at the present time, t, from the knowledge of \tilde{x} in the immediate past, t-dt. Of course we should expect the approximate solution, \tilde{x}, to approach the exact solution, x, as the time step dt, approaches zero. A careful explication of the matrix exponential and its relationship to the transfer function will have to wait until we have mastered the inverse laplace transform.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/07%3A_Spectral_Theory/7.03%3A_Applications_of_Spectral_TheorySuppose we have a matrix A and we want to find A50 . One could try to multiply A with itself 50 times, but this is computationally extremely intensive (try it!). However diagonalization allows us...Suppose we have a matrix A and we want to find A50 . One could try to multiply A with itself 50 times, but this is computationally extremely intensive (try it!). However diagonalization allows us to compute high powers of a matrix relatively easily.