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Mathematics-Online course: Linear Algebra - Normal Forms - Eigenvalues and Eigenvectors

Sum and Product of Eigenvalues

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Let $ \lambda_i$ denote the $ n$ not necessarily different eigenvalues of $ n\times n$ matrix $ A$. Then the follwing holds true:

$\displaystyle \sum_{i=1}^n \lambda_i = \operatorname{Sp}(A),\quad \prod_{i=1}^n \lambda_i = \operatorname{det}(A) \,. $

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The matrix

$\displaystyle A=\left(\begin{array}{rr}a & b\\ c & d\end{array}\right) $

has the characteristic polynomial

$\displaystyle p_A(\lambda)=\lambda^2-(a+d)\lambda+(ad-bc) $

with zeros

$\displaystyle \lambda_{1,2}=\frac{(a+d)\pm\sqrt{(a+d)^2-4(ad-bc)}}{2}\,. $

Forming the sum of the eigenvalues the root expression vanishes and we obtain

$\displaystyle \lambda_1+\lambda_2=\frac{(a+d)+(a+d)}{2}=a+d=\operatorname{Spur}(A)\,. $

Multiplying the two eigenvalues we get by the third binomial formula:

$\displaystyle \lambda_1 \lambda_2=\frac{(a+d)^2-(a+d)^2+4(ad-bc)}{4}=ad-bc=\operatorname{det}(A)\,. $

  automatically generated 4/21/2005