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Eigenvalues in 2D

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Exercise

https://texercises.com/exercise/eigenvalues-in-2d/

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Exercise:
abcliste abc Show that the eigenvalues of a real times matrix bf A leftmatrixa & b c & dmatrixright are given by lambda_ fractaupm sqrttau^-Delta where taua+d is the trace and Deltaad-bc is the determinant of the matrix. abc Find and example of a matrix for which the eigenvalues are purely imaginary. abcliste

Solution:
abcliste abc Eigenvalues are characterised by bf A bf v lambda bf v This can be written more explicitely as leftmatrixa & b c & dmatrixright leftmatrixv_ v_matrixright lambda leftmatrixv_ v_matrixright or leftmatrixa-lambda & b c & d-lambdamatrixright leftmatrixv_ v_matrixright For non-trivial solutions i.e. v_neq or v_neq the determinant of the matrix in the above line has to be zero: det leftmatrixa-lambda & b c & d-lambdamatrixright Using the expression for the determinant of a times matrix we find a-lambdad-lambda-bc lambda^-a+dlambda+ad-bc lambda^-taulambda+Delta The solutions are given by the well-known formula for quadratic s. abc For purely imaginary eigenvalues the trace has to be zero a+d and the determinant has to be positive: ad-bc a-a-bc -a^-bc It follows that b and c have to have different signs and their product has to be greater than a^. A simple example would be the matrix leftmatrix & - & -matrixright with eigenvalues pm i. abcliste

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Branches	Linear Algebra
Tags	eigenvalue, matrix
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Difficulty	(1, default)
Points	0 (default)
Language	(English)
Type	Calculative / Quantity
Creator	by
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