What is the relation between trace and determinant?
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The determinant and the trace are two quite different beasts, little relation can be found among them.
We know tr(M)=∑λitr(M)=∑λi and det(M)=∏λidet(M)=∏λi, where λiλi are the eigenvalues.
If the matrix is not only symmetric (hermitic) but also positive definite, then the eigenvalues are real and positive. Then, using the AM GM inequality, we get
tr(M)n≥det(M)1/ntr(M)n≥det(M)1/n
(equality holds iff M=λIM=λI for some λ≥0λ≥0)
We know tr(M)=∑λitr(M)=∑λi and det(M)=∏λidet(M)=∏λi, where λiλi are the eigenvalues.
If the matrix is not only symmetric (hermitic) but also positive definite, then the eigenvalues are real and positive. Then, using the AM GM inequality, we get
tr(M)n≥det(M)1/ntr(M)n≥det(M)1/n
(equality holds iff M=λIM=λI for some λ≥0λ≥0)
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