Question

Let \(A\) be an invertible \(n \times n\) matrix. a. Show that the eigenvalues of \(A\) are nonzero. b. Show that the eigenvalues of \(A^{-1}\) are precisely the numbers \(1 / \lambda,\) where \(\lambda\) is an eigenvalue of \(A\). c. Show that \(c_{A^{-1}}(x)=\frac{(-x)^{n}}{\operatorname{det} A} c_{A}\left(\frac{1}{x}\right)\).

Short answer

Eigenvalues of \( A \) are nonzero, eigenvalues of \( A^{-1} \) are \( 1/\lambda \), and the characteristic polynomials confirm this relationship.

Step-by-step solution

Understanding Eigenvalues

An eigenvalue \( \lambda \) for a matrix \( A \) is a scalar value that satisfies the equation \( A\mathbf{v} = \lambda \mathbf{v} \), where \( \mathbf{v} \) is a non-zero vector known as an eigenvector.

Prove Eigenvalues of A are Nonzero

Since \( A \) is invertible, its determinant is nonzero, \( \det(A) eq 0 \). If \( \lambda = 0 \) were an eigenvalue, then there would exist a non-zero vector \( \mathbf{v} \) such that \( A \mathbf{v} = 0 \), contradicting \( A \) being invertible. Therefore, all eigenvalues \( \lambda \) of \( A \) must be nonzero.

Find Eigenvalues of the Inverse Matrix

Suppose \( \lambda \) is an eigenvalue of \( A \), then \( A \mathbf{v} = \lambda \mathbf{v} \) for some vector \( \mathbf{v} \). Applying \( A^{-1} \) to both sides, \( A^{-1} A \mathbf{v} = A^{-1}(\lambda \mathbf{v}) \) simplifies to \( \mathbf{v} = \lambda A^{-1} \mathbf{v} \), or equivalently, \( A^{-1} \mathbf{v} = \frac{1}{\lambda} \mathbf{v} \). Hence, \( \frac{1}{\lambda} \) is an eigenvalue of \( A^{-1} \).

Characteristic Polynomial of A and A^{-1}

The characteristic polynomial of \( A \), \( c_A(x) = \det(xI - A) \), expands to \( \prod_{i=1}^{n} (x - \lambda_i) \), where \( \lambda_i \) are the eigenvalues of \( A \). The polynomial for \( A^{-1} \) is \( c_{A^{-1}}(x) = \det(xI - A^{-1}) \).

Transform Polynomial of A^{-1}

For the inverse matrix \( A^{-1} \), the transformation \( xI - A^{-1} = A^{-1}(xA - I) \) and taking determinants gives \[ \det(xI - A^{-1}) = \frac{1}{\det(A)} \det(xA - I)\].

Substitute and Rearrange

Substitute \( x = \frac{1}{y} \) in \( \det(yI - A) = \prod_{i=1}^{n} (y - \lambda_i) \), giving \( y = \frac{1}{x} \). This becomes \( \det(xI - A^{-1}) = \frac{(-1)^{n}x^{n}}{\det A} \det(I - xA^{-1}) = \frac{(-x)^{n}}{\det A} c_A(\frac{1}{x}) \).

Conclusion

The eigenvalues of an invertible matrix \( A \) are nonzero. The eigenvalues of \( A^{-1} \) are \( 1/\lambda \). The characteristic polynomial confirms \( c_{A^{-1}}(x) = \frac{(-x)^{n}}{\det A} c_A\left(\frac{1}{x}\right) \).

Key concepts

Eigenvalues

Eigenvalues are crucial in understanding the behavior of matrices in linear algebra. An eigenvalue of a matrix \( A \) is a scalar, \( \lambda \), such that there exists a non-zero vector \( \mathbf{v} \) (the eigenvector), satisfying the equation \( A\mathbf{v} = \lambda \mathbf{v} \). This essentially means that applying matrix \( A \) to vector \( \mathbf{v} \) stretches or compresses it by the factor \( \lambda \), without changing its direction. In the context of invertible matrices, all eigenvalues must be non-zero because a zero eigenvalue would imply the existence of a vector \( \mathbf{v} \) for which \( A\mathbf{v} = 0 \), contradicting the definition of invertibility, which requires a full rank and non-zero determinant. Thus, invertible matrices have eigenvalues that are all non-zero, ensuring that no such vector is nullified by the matrix.

Invertible Matrix

An invertible matrix, often referred to as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. Only square matrices can possibly be invertible. For a matrix \( A \) to be invertible, there must exist a matrix \( A^{-1} \) such that the multiplication of \( A \) and \( A^{-1} \) yields the identity matrix \( I \), i.e., \( AA^{-1} = A^{-1}A = I \). The existence of an inverse means \( \det(A) eq 0 \), ensuring that the matrix can be reversed or undone. This concept is essential in solving linear systems and performing various matrix operations where reversing the transformation represented by the matrix is necessary. Furthermore, the properties of invertible matrices extend to the fact that their eigenvalues cannot be zero, as established earlier.

Characteristic Polynomial

The characteristic polynomial of a matrix \( A \) provides a polynomial equation whose roots are the eigenvalues of the matrix. Given a matrix \( A \), the characteristic polynomial \( c_A(x) \) is defined by \( \det(xI - A) \), where \( I \) is the identity matrix of the same size as \( A \). Expanding this determinant typically results in a polynomial of degree \( n \) if \( A \) is \( n \times n \). Each root of this polynomial corresponds to an eigenvalue of \( A \). For instance, if the characteristic polynomial is \( (x - \lambda_1)(x - \lambda_2) \ldots (x - \lambda_n) \), the values \( \lambda_1, \lambda_2, \ldots, \lambda_n \) are the eigenvalues. This polynomial is not just a random mathematical construct; it forms the basis for understanding how the matrix scales space, providing critical insight into the stability and behavior of systems, particularly in differential equations and dynamic systems.

Matrix Inverse

The inverse of a matrix \( A \) is another matrix \( A^{-1} \) such that when multiplied, they yield the identity matrix, i.e., \( AA^{-1} = I \) and \( A^{-1}A = I \). Not every matrix has an inverse; matrices that do are called invertible. To find the inverse of a matrix, it is essential to perform operations that conform to specific rules, often involving the adjugate (or adjoint) of the matrix and the determinant. The process can be intricate for higher-dimension matrices but is pivotal for solving linear systems where you need to isolate variables or revert transformations. The eigenvalues of \( A^{-1} \) are particularly interesting because they are simply the reciprocals of the eigenvalues of \( A \). For instance, if \( \lambda \) is an eigenvalue of \( A \), then \( \frac{1}{\lambda} \) is an eigenvalue of \( A^{-1} \). This allows transformational insights, enabling reverse engineering of matrix transformations applicable in various fields such as physics, computer graphics, and engineering.