Question

Let \(A=\left[\begin{array}{lll}0 & a & b \\ a & 0 & c \\ b & c & 0\end{array}\right]\) and \(B=\left[\begin{array}{lll}c & a & b \\ a & b & c \\ b & c & a\end{array}\right]\) a. Show that \(x^{3}-\left(a^{2}+b^{2}+c^{2}\right) x-2 a b c\) has real roots by considering \(A\). b. Show that \(a^{2}+b^{2}+c^{2} \geq a b+a c+b c\) by considering \(B\).

Short answer

The polynomial has real roots, and the inequality holds.

Step-by-step solution

Compute Eigenvalues of Matrix A

The given matrix \( A \) is \( \begin{pmatrix} 0 & a & b \ a & 0 & c \ b & c & 0 \end{pmatrix} \). To find if its characteristic polynomial \( x^3 - (a^2+b^2+c^2)x - 2abc \) has real roots, we compute the eigenvalues by noting structure of \( A \) and its symmetry. We observe that this matrix represents a special case of a symmetric matrix which can be real because real symmetric matrices have real eigenvalues.

Check for Positive Definiteness

The characteristic polynomial \( x^3 - (a^2+b^2+c^2)x - 2abc \) is derived from the determinant of \( A-xI \) where \( I \) is the identity matrix. For such polynomials derived from real matrices, the real nature of eigenvalues is checked using specific parameter values which affect the signs of coefficients. The coefficients simply need to reflect reality from symmetric properties to guarantee roots in reals.

Apply Cauchy-Schwarz Inequality to Matrix B

For matrix \( B = \begin{pmatrix} c & a & b \ a & b & c \ b & c & a \end{pmatrix} \), consider the inequality \( a^2+b^2+c^2 \geq ab+ac+bc \). This is a direct application of Cauchy-Schwarz, given any real numbers or real-valued matrices the sum of squared components will dominate any linear product combinations, deriving from vector norms.

Synthesize Conclusions

We conclude by recognizing under symmetry and structure, matrix \( A \)'s polynomial derives from characteristics assuring real roots through symmetric real-valued algebraic expressions, and matrix \( B \)'s inequality follows from algebraic theory supporting non-negativity and symmetry properties like those in vector/matrix norms. Thus the conditions hold with provided reasoning steps.

Key concepts

Eigenvalues

Eigenvalues are special scalars associated with a square matrix that provide profound insights into the matrix's characteristics. They are essentially the solutions to the characteristic equation of the matrix. When we have a matrix like \( A \), which is \( \begin{pmatrix} 0 & a & b \ a & 0 & c \ b & c & 0 \end{pmatrix} \), its eigenvalues can be found by solving the equation that emerges from setting \( \det(A - \lambda I) = 0 \), where \( \lambda \) represents the eigenvalues and \( I \) is the identity matrix.

For symmetric matrices such as \( A \), all eigenvalues are real. This property is significantly useful because it provides a guarantee about the type of roots you will find when solving its characteristic polynomial. Real symmetric matrices always have real eigenvalues, which reassures us that the associated polynomial \( x^3 - (a^2 + b^2 + c^2)x - 2abc \) indeed has real roots. Thus, in studying eigenvalues, we delve into understanding both the matrix's structure and the characteristics of these special scalars.

Characteristic Polynomial

The characteristic polynomial of a matrix is a key algebraic expression that encodes important information about the matrix. It is derived from the determinant of \( A - xI \), where \( A \) is the matrix in question and \( I \) is the identity matrix. The zeros of this polynomial are the eigenvalues of the matrix, which makes it an essential tool for understanding a matrix's properties.

Taking matrix \( A \) as an example, its characteristic polynomial is \( x^3 - (a^2 + b^2 + c^2)x - 2abc \). This polynomial is constructed by equating the determinant \( \det(A - xI) \) to zero. The form of this polynomial provides insight into the matrix's eigenvalues and their nature (real or complex).

Observing the coefficients of the polynomial, particularly the terms \( a^2 + b^2 + c^2 \) and \( 2abc \), gives additional information about the matrix characteristics. The polynomial serves as a bridge, connecting the matrix's algebraic properties with its geometric interpretation.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a fundamental inequality in mathematics that holds for any real-valued numbers and vectors. It's recognized for its robust role in many areas, including algebra, calculus, and linear algebra. The inequality states that for any vectors \( \mathbf{u} \) and \( \mathbf{v} \), the absolute value of their dot product is less than or equal to the product of the magnitudes of the vectors: \[ |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\| \]In the context of matrix \( B \), \[ B = \begin{pmatrix} c & a & b \ a & b & c \ b & c & a \end{pmatrix} \], we can relate it to the inequality \( a^2 + b^2 + c^2 \geq ab + ac + bc \). This inequality stems from applying Cauchy-Schwarz to vectors formed from the components of \( B \).

The rationale here is rooted in the fact that the sum of squares \( a^2 + b^2 + c^2 \) is always a stronger measure than the sum of pairwise products \( ab + ac + bc \). This principle is a manifestation of Cauchy-Schwarz, showing that the algebraic structure ensures certain dominance over simpler linear combinations.

Symmetric Matrices

Symmetric matrices are a special class of square matrices that exhibit appealing properties. A symmetric matrix is one where its transpose is equal to itself. That means if \( A \) is symmetric, then \( A = A^T \).

One of the significant traits of symmetric matrices is that their eigenvalues are always real numbers. This is a consequence of the fact that they can always be diagonalized by an orthogonal matrix. Symmetric matrices also have eigenvectors that are orthogonal to each other, making them very useful in various applications like principal component analysis and quantum mechanics.

In our example, matrices \( A \) and \( B \) both have symmetric patterns. The matrix \( A \), given as \( \begin{pmatrix} 0 & a & b \ a & 0 & c \ b & c & 0 \end{pmatrix} \), exhibits symmetry across its diagonal. As a result, when solving for eigenvalues, we can rest assured they will be real. Symmetric matrices, through their properties, simplify many complex computations and enable deeper insights into matrix characteristics and behaviors.