Expression 10: lambda Subscript, 1 , Baseline equals StartFraction, "a" Subscript, 11 , Baseline plus "a" Subscript, 22 , Baseline plus StartRoot, left parenthesis, "a" Subscript, 11 , Baseline plus "a" Subscript, 22 , Baseline , right parenthesis squared minus 4 left parenthesis, "a" Subscript, 11 , Baseline "a" Subscript, 22 , Baseline minus "a" Subscript, 12 , Baseline "a" Subscript, 21 , Baseline , right parenthesis , EndRoot Over 2 , EndFractionλ1=a11+a22+a11+a222−4a11a22−a12a212
equals=
22
10
Expression 11: lambda Subscript, 2 , Baseline equals StartFraction, "a" Subscript, 11 , Baseline plus "a" Subscript, 22 , Baseline minus StartRoot, left parenthesis, "a" Subscript, 11 , Baseline plus "a" Subscript, 22 , Baseline , right parenthesis squared minus 4 left parenthesis, "a" Subscript, 11 , Baseline "a" Subscript, 22 , Baseline minus "a" Subscript, 12 , Baseline "a" Subscript, 21 , Baseline , right parenthesis , EndRoot Over 2 , EndFractionλ2=a11+a22−a11+a222−4a11a22−a12a212
equals=
11
11
Black lines/dots: a collection of vectors that fills out a unit square (pre-transformation)
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Orange line: the line of all eigenvectors associated with the FIRST eigenvalue
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14
Green line: the line of all eigenvectors associated with the SECOND eigenvalue
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16
Press play (or slide c from 0 to 1) to animate the process of transforming the vectors which fill out the unit square by the matrix A. Notice which vectors change orientation, and which ones just scale.