Transcription of Chapter 6 Eigenvalues and Eigenvectors - MIT Mathematics
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Chapter 6 Eigenvalues and Eigenvectors - MIT Mathematics
Chapter 6 Eigenvalues and Introduction to Eigenvalues '&$%1 Aneigenvectorxlies along the same line asAx:Ax= .2 IfAx= xthenA2x= 2xandA 1x= 1xand(A+cI)x= ( +c)x: the xthen(A I)x=0andA Iis singular anddet(A I)= s bydetA= ( 1)( 2) ( n)and diagonal suma11+a22+ +ann=sum of have =1and0. Reflections have1and 1. Rotations haveei ande i :complex!This Chapter enters a new part of linear algebra. The first part was aboutAx=b:balance and equilibrium and steady state. Now the second part is aboutchange. Timeenters the picture continuous time in a differential equationdu/dt=Auor time stepsin a difference equationuk+1=Auk. Those equations are NOT solved by key idea is to avoid all the complications presented by the matrixA.
1. Markov matrix: Each column of P adds to 1, so λ = 1 is an eigenvalue. 2. P is singular, so λ = 0 is an eigenvalue. 3. P is symmetric, so its eigenvectors (1,1) and (1,−1) are perpendicular. The only eigenvalues of a projection matrix are 0and 1. The eigenvectors for λ = 0(which means Px = 0x)fill up the nullspace.
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