Study Notes

by Nasser Abbasi

Notes on Linear algebra

1.     If M is a symmetric matrix then C M (C^T)is a diagonal matrix, where C is a matrix whose rows are the eigen vectors of M. example

Remove["Global`*"]
m=(({{1, 4, 5}, {4, 8, 9}, {5, 9, 4}}));
v = Eigenvectors[m];

Chop[N[v . m . Transpose[v]]] // 
      MatrixForm

2    If M is a symmetric matrix then (e^M)=(C^T)(({{(e^((Subscript[λ, 1]) t)), 0, 0}, {0, (e^((Subscript[λ, 2]) t)), 0}, {0, 0, (e^((Subscript[λ, n]) t))}}))C where (Subscript[λ, i]) are the eigenvalues of M and C is a matrix whose rows are the eigen vectors of M. example

Remove["Global`*"]
badMatrix=(({{1, 4, 5}, {4, 8, 7}, {5, 7, 4}}));
goodMatrix=(({{0.624961238712114`, 0.9156593589568051`, 0.9594314541903537`}, {0.9156593589568051`, 0.9588324892854148`, 0.6159355337104255`}, {0.9594314541903537`, 0.6159355337104255`, 1.313001322953736`}}));
m = badMatrix; 
{λ, v} = 
        Eigensystem[m];
Chop[
        N[Transpose[v] . DiagonalMatrix[(E^λ)] . v]] // 
      MatrixForm

Now evaluate (e^M)  using Mathematica function

(N[MatrixExp[m ]]) // MatrixForm