Fun fact for a 1x1 Matrix both are the same
The determinant of an 𝑛×𝑛 matrix is equal to the 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 of its eigenvalues The trace is equal to the 𝑠𝑢𝑚 of its eigenvaluespic.twitter.com/sWdUWUeP0y
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What is a trace?
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The trace of a matrix is usually defined to be the sum of its diagonal entries.
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Provided that the matrix M is diagonazible with diagonal matrix D M=S^(-1)D S detM=det(S^(-1)D S)= det(D S S^(-1))=detD="product of eigenvalues" (using detAB=detBA) trM=tr(S^(-1)D S)= tr(DS S^(-1))=trD= "sum of eigenvalues" (using trAB=trBA, cyclic property of the trace).
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Yes but it holds more generally!
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Fun fact: all these are examples of elementary symmetric polynomials! https://twitter.com/InertialObservr/status/1176568421242363905 …pic.twitter.com/zaCQyvDdAg
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Read more: https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial … Why should we care about elementary symmetric polynomials of the eigenvalues of a matrix? Because of the Cayley-Hamilton theorem: every matrix satisfies its own characteristic polynomial!! https://en.wikipedia.org/wiki/Cayley–Hamilton_theorem …
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The determinant of a matrix also represents the volume of the n-dimensional parallelepiped created by its column vectors!pic.twitter.com/nsuBBQvbPB
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