Matrix Operations This is just a warmup for solutions of the eigenmodes of the one dimensional Schrodinger equation, It includes all the standard square matrix functions like multiplication, determinant, and inversion. The learner who wants to use these function for themselves is welcome to hit F12 on any Windows 10 computer and copy them into their own Javascript program. Or just change the values of the list a or matrix m in the reSet function in this program and re-run the program by hitting F5. Note that the only matrices cosidered here are square (nxn) matrices. The program culminates with solutions for eigenvalues and eigenvalues of a 3x3 matrix but the functions available could do the same for much larger matrices. In physics second order differential equations are extremely common. These can be solved approximately by forming what is called a tridiagonal matrix where the only non-zero elements are the diagonal ones and the two next to diagonal ones next to the diagonal. Often the neighboring elements are constants which makes the problem even simpler, for exanple |d1 b 0 0 0 ||v1| |v1| |b d2 b 0 0 ||v2| |v2| |0 b d3 b 0 ||v3|=lambda|v3| |0 0 b d4 b ||v4| |v4| |0 0 0 b d5||v5| |v5| where lambda is the eigenvalue and the v vector is the eigenvector of the matrix on the left. Another use of this tridiagonal matrix is to solve for the values v given its initial conditions. This can be done by using Newton's method as suggested below. In the equations below, f(v) is the functional part of the diagonal of the matrix. F′(v) = tridiag (1/∆x2 , −2/∆x2 + ∂f ∂vi , 1/∆x2) F(vk)==F(vk)+F'(vk)(v-vk) v-vk==-(Inverse)(F')*F(vk) Invert F' to find v-vk and then repeat with the new vk.
Click Here for Power Method Equations Click Here for Inverse Power Method Equations Click Here for Eigenvector Equations