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To fix notation, let A be an NX N-matrix, and let us recall the characteristic polynomial of A: PA(x) = det(A - x· Id) = c₁(A) — Cn-1(A)x + ··· + c₁ (A) · (-x)"−1 + (-x)", where the ck (A) E F are coefficients of the polynomial. Exercise A (see below for markdown cell) Show that c, (A) = det(A) and c₁ (A) = tr(A).

To fix notation, let A be an NX N-matrix, and let us recall the characteristic polynomial of A: PA(x) = det(A - x· Id) = c₁(A) — Cn-1(A)x + ··· + c₁ (A) · (-x)"−1 + (-x)", where the ck (A) E F are coefficients of the polynomial. Exercise A (see below for markdown cell) Show that c, (A) = det(A) and c₁ (A) = tr(A).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
In the notes, you've encountered permutation matrices and seen that the determinant of a permutation matrix is equal to the signature of the permutation. In this exploration, you'll consider the entire characteristic polynomial of a permutation matrix (of which the determinant is just its constant coefficient). To fix notation, let A be an N X N-matrix, and let us recall the characteristic polynomial of A: PA(x) = det(A − x · Id) = c₂(A) − Cn−1(A)x + + c₁ (A) · (−x)¹−¹ + (−x)”, where the ck(A) E F are coefficients of the polynomial. Exercise A (see below for markdown cell) Show that c₁(A) = det(A) and c₁ (A) = tr(A).
Transcribed Image Text:In the notes, you've encountered permutation matrices and seen that the determinant of a permutation matrix is equal to the signature of the permutation. In this exploration, you'll consider the entire characteristic polynomial of a permutation matrix (of which the determinant is just its constant coefficient). To fix notation, let A be an N X N-matrix, and let us recall the characteristic polynomial of A: PA(x) = det(A − x · Id) = c₂(A) − Cn−1(A)x + + c₁ (A) · (−x)¹−¹ + (−x)”, where the ck(A) E F are coefficients of the polynomial. Exercise A (see below for markdown cell) Show that c₁(A) = det(A) and c₁ (A) = tr(A).
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