The matrix A = [2 7 −1; 0 −3 4; 0 0 5] (rows separated by semicolons) has eigenvalues λ₁, λ₂ and λ₃. Find λ₁² + λ₂² + λ₃².

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Squared eigenvalues

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The matrix $A = 200 7 - 3 0 - 1 45$ (rows separated by semicolons) has eigenvalues $λ_{1}$ , $λ_{2}$ and $λ_{3}$ . Find $λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2}$ .

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Hint

Look at the shape of the matrix before you write out a characteristic polynomial.

Solution

The matrix is upper triangular, and so is $A - λ I$ , so $det (A - λ I)$ is the product of the diagonal entries of $A - λ I$ .
That product is $(2 - λ) (- 3 - λ) (5 - λ)$ , so the eigenvalues are the diagonal entries $2$ , $- 3$ and $5$ .
Their squares sum to $4 + 9 + 25 = 38$ .

Answer: $38$

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