Pamela Leutwyler. Find the eigenvalues and eigenvectors next.

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Presentation transcript:

Pamela Leutwyler

Find the eigenvalues and eigenvectors next

next

next

next

next

next

next

next

next characteristic polynomial

next characteristic polynomial

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next potential rational roots:1,-1,3,-3,9,-9 synthetic division: This is not zero. 1 is not a root.

next potential rational roots:1,-1,3,-3,9,-9 synthetic division:

next synthetic division: potential rational roots:1,-1,3,-3,9,-9

next synthetic division: potential rational roots:1,-1,3,-3,9,-9

next synthetic division: potential rational roots:1,-1,3,-3,9,-9

next synthetic division: potential rational roots:1,-1,3,-3,9,

next synthetic division: potential rational roots:1,-1,3,-3,9, This is zero. -3 is a root.

next synthetic division: potential rational roots:1,-1,3,-3,9,

next synthetic division: potential rational roots:1,-1,3,-3,9,

next synthetic division: The eigenvalues are: -3, -3, -1

next The eigenvalues are: -3, -3, -1 To find an eigenvector belonging to the repeated root –3, consider the null space of the matrix –3I - A

next The eigenvalues are: -3, -3, -1 To find an eigenvector belonging to the repeated root –3, consider the null space of the matrix –3I - A The 2 dimensional null space of this matrix has basis =

next The eigenvalues are: -3, -3, -1 To find an eigenvector belonging to the repeated root –1, consider the null space of the matrix –1I - A The null space of this matrix has basis =

next The eigenvalues are: -3, -3, -1 The eigenvectors are:

next The eigenvalues are: -3, -3, -1 The eigenvectors are:

The eigenvalues are: -3, -3, -1 The eigenvectors are: A P P –1 diagonal matrix that is similar to A