Study Notes

by Nasser Abbasi

Notes on Linear algebra

1.     If M is a symmetric matrix then C M C^Tis 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

( {{48.2835, 0, 0}, {0, -6.37096, 0}, {0, 0, -29.445}} )

2    If M is a symmetric matrix then ^M=C^T(

^(λ_1t) 0 0
0 ^(λ_2t) 0
0 0 ^(λ_nt)
)C where 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 = ( {{0.624961, 0.915659, 0.959431}, {0.915659, 0.958832, 0.615936}, {0.959431, 0.615936, 1.313}} ) ;

m = badMatrix ;

{λ, v} = Eigensystem[m] ;

Chop[N[Transpose[v] . DiagonalMatrix[^λ] . v]]//MatrixForm

( {{3.51857*10^6, 6.48044*10^6, 5.29451*10^6}, {6.48044*10^6, 1.19356*10^7, 9.75134*10^6}, {5.29451*10^6, 9.75134*10^6, 7.96683*10^6}} )

Now evaluate ^M  using Mathematica function

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

( {{1.19687*10^6, 2.20437*10^6, 1.80097*10^6}, {2.20437*10^6, 4.05998*10^6, 3.317*10^6}, {1.80097*10^6, 3.317*10^6, 2.70998*10^6}} )


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