Question

We are given the \(n\)-th order system \(\dot{x}=A x\) with $$ A=\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & 0 & 1 \\ -a_{0} & -a_{1} & \cdots & -a_{n-2} & -a_{n-1} \end{array}\right) $$ Show that the chanacteristic polynomial of \(A\) is $$ \lambda^{n}+a_{n-1} \lambda^{n-1}+\ldots+a_{1} \lambda+a_{0} $$ If \(\lambda\) is an eigenvalue of \(A\), then prove that the corresponding eigenvector is $$ \left(1, \lambda, \lambda^{2}, \ldots, \lambda^{n-1}\right)^{T} $$

Short answer

The characteristic polynomial is \( \lambda^n + a_{n-1} \lambda^{n-1} + \ldots + a_0 \), and an eigenvector is \( (1, \lambda, \lambda^2, \ldots, \lambda^{n-1})^T \).

Step-by-step solution

Form the Characteristic Polynomial

The characteristic polynomial of a matrix \( A \) is given by \( \det(\lambda I - A) \), where \( I \) is the identity matrix of the same size as \( A \). For this system, \( A \) is a companion matrix of the polynomial. Specifically, \( (\lambda I - A) \) will have terms directly contributing to forming the polynomial \( \lambda^n + a_{n-1} \lambda^{n-1} + ... + a_0 \). Constructing these, we see the diagonal entries as \( \lambda \) and all other non-zero entries as elements of \( A \), resulting in the polynomial as given.

Express Determinant in Terms of Coefficients

The matrix \( \lambda I - A \) has the form of an upper triangular matrix with \( \lambda \) terms on the diagonal. The determinant of an upper triangular matrix is the product of its diagonal elements: \( \lambda \times \lambda \times \ldots \times \lambda = \lambda^n \). Each anti-diagonal contributes coefficients \( a_i \) pulled from \( -A \). Expanding the determinant mirrors expanding \( (\lambda - \lambda_1)(\lambda - \lambda_2)\ldots(\lambda - \lambda_n) \) as per the characteristic polynomial definition.

Identify Eigenvalues of Matrix A

The characteristic polynomial of \( A \) derived in Step 1, \( \lambda^n + a_{n-1} \lambda^{n-1} + \ldots + a_0 \), when set to zero gives the eigenvalues of \( A \). These \( \lambda_i \)'s are the roots of this characteristic equation.

Find Eigenvectors Corresponding to Each Eigenvalue

To find the eigenvector corresponding to an eigenvalue \( \lambda \), solve \((A - \lambda I) \mathbf{v} = \mathbf{0} \), where \( \mathbf{v} \) is the eigenvector. For a companion matrix, this resolves to finding solutions \( \mathbf{v} = \left(1, \lambda, \lambda^2, \ldots, \lambda^{n-1}\right)^T \) to the above differential equation system, as observed from recursive expansion nature of matrix polynomial relationships.

Key concepts

Eigenvalues

In the context of linear algebra, eigenvalues play a crucial role in understanding the behavior of linear transformations. An eigenvalue is a special number associated with a matrix that gives insight into the matrix's structural properties and dynamics. For a given square matrix \(A\), an eigenvalue \(\lambda\) satisfies the equation \((A - \lambda I)\mathbf{v} = \mathbf{0}\), where \(I\) is the identity matrix, and \(\mathbf{v}\) is a non-zero vector known as the eigenvector.
When we solve this equation, we are essentially finding the \(\lambda\)'s that make this equation hold true, meaning the transformation \(A\) stretches or shrinks \(\mathbf{v}\) by a factor of \(\lambda\) without changing its direction. In our exercise, these are found by solving the characteristic equation \(\lambda^n + a_{n-1} \lambda^{n-1} + ... + a_0 = 0\).

• Eigenvalues help analyze matrices for stability and transformation properties.
• They are solutions to the characteristic polynomial of a matrix.
• In our system, they help describe the dynamic behavior of the state system.

Eigenvectors

Eigenvectors are essential paired concepts with eigenvalues, providing a vector perspective on how transformations act over spaces. An eigenvector of a matrix \(A\) corresponding to an eigenvalue \(\lambda\) is a non-zero vector that satisfies \((A - \lambda I)\mathbf{v} = \mathbf{0}\).
In simpler terms, this vector \(\mathbf{v}\) does not change direction when the linear transformation by \(A\) is applied to it. Instead, it is merely scaled by the factor \(\lambda\). For the exercise at hand, if \(\lambda\) is an eigenvalue, the corresponding eigenvector can be expressed as \((1, \lambda, \lambda^2, \ldots, \lambda^{n-1})^T\).

• These vectors determine invariant directions in linear transformations.
• Important for applications like stability analysis and vibrations in mechanical systems.
• In our problem, eigenvectors demonstrate how states multiply recursively.

Companion Matrix

The concept of a companion matrix is intertwined with polynomials, as it represents a tool to easily derive the characteristic polynomial. For a polynomial like \(\lambda^n + a_{n-1} \lambda^{n-1} + \ldots + a_0\), the associated companion matrix \(A\) is structured to help us directly draw this polynomial from its determinant.
The companion matrix for the given polynomial starts with zeroes and an identity sequence that systematically includes coefficients \(-a_i\) along its last row. This configuration ensures that when forming \(\lambda I - A\), the characteristics of the polynomial are maintained. In particular, it effectively encodes the polynomial's coefficients.

• A key role of companion matrices is facilitating polynomial evaluations.
• These matrices are square and structured to reflect polynomial linear expansions.
• In this task, they simplify deriving the characteristic polynomial from matrix properties.

Determinant

Determinants offer a scalar metric that provides insight into matrices. Specifically, it helps determine linear dependences, volumes in linear transformations, and solvability of systems of linear equations. The determinant of a matrix \(A\) is denoted as \( \det(A) \) and mathematically formed from the product of its diagonal entries when \(A\) is in triangular form.
In the context of eigenvalues, the determinant of \(\lambda I - A\) gives us the characteristic polynomial. For our system, the matrix \(\lambda I - A\) is upper triangular, leading its determinant to be the product of \(\lambda\)'s on the diagonal, generating \(\lambda^n\) and incorporating terms that mirror the polynomial coefficients. This setup shows how determinant properties lead us to characteristic equations in systemic approaches.

• Determinants help verify matrix invertibility and eigenvalue evaluations.
• They link matrix properties to algebraic expressions in a functional manner.
• In this scenario, they reflect the construction of a characteristic equation tied to transformation properties.