(a) Find the explicit formula for the characteristic polynomial det(A-λI)=-λ^3+a λ^2-b λ+c of a

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(a) Find the explicit formula for the characteristic...

(a) Find the explicit formula for the characteristic polynomial $\operatorname{det}(A-\lambda \mathrm{I})=-\lambda^3+a \lambda^2-b \lambda+c$ of a general $3 \times 3$ matrix. Verify that $a=\operatorname{tr} A$, $c=\operatorname{det} A$. What is the formula for $b$ ? (b) Prove that if $A$ has eigenvalues $\lambda_1, \lambda_2, \lambda_3$, then $a=\operatorname{tr} A=\lambda_1+\lambda_2+\lambda_3, b=\lambda_1 \lambda_2+\lambda_1 \lambda_3+\lambda_2 \lambda_3, c=\operatorname{det} A=\lambda_1 \lambda_2 \lambda_3$.

Key Concepts

Elementary Symmetric Polynomials in Eigenvalues

The coefficients of the characteristic polynomial are intimately connected with the elementary symmetric polynomials of the eigenvalues. For a 3×3 matrix, the first elementary symmetric polynomial is the sum of eigenvalues (equal to the trace, a), the second is the sum of the products of the eigenvalues taken two at a time (coefficient b), and the third is the product of all three eigenvalues (equal to the determinant, c). This relationship provides a powerful bridge between a matrix’s algebraic properties and its spectral characteristics.

Principal Minors and the Coefficient b

Coefficient b in the characteristic polynomial of a 3×3 matrix is obtained by summing all the 2×2 principal minors, which are the determinants of the submatrices produced by removing one row and one column. This sum represents the combined effects of all possible pairwise interactions among the matrix elements and is also equivalent to the sum of the products of eigenvalue pairs. Thus, b serves as a measure of the intermediate spectral behavior of the matrix.

Trace of a Matrix

The trace of a matrix is the sum of its diagonal elements and is invariant under change of basis. In the context of the characteristic polynomial of a 3×3 matrix, the coefficient a is equal to the trace of the matrix, which is also the sum of its eigenvalues. This property establishes a direct link between the matrix’s internal structure and its spectral characteristics.

Determinant of a Matrix

The determinant is a scalar attribute that describes how a matrix scales volume when it represents a linear transformation. It appears as the constant term in the characteristic polynomial of a 3×3 matrix (up to sign) and is equal to the product of the eigenvalues. In the given polynomial, c corresponds to the determinant, reflecting another intrinsic invariant of the matrix.

Characteristic Polynomial

The characteristic polynomial of a matrix is defined as det(A - ?I) and its roots are the eigenvalues of A. For a 3×3 matrix, it can be written in the form –?³ + a?² – b? + c, where the coefficients a, b, and c encapsulate key matrix invariants. This polynomial provides a compact algebraic formulation that connects a matrix's numerical properties with its eigenvalues.

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