Question

By direct computation, show that there is no invertible complex matrix \(C\) such that $$ C^{-1}\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right] C=\left[\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$

Short answer

No invertible complex matrix \( C \) exists because \( A \) and \( B \) have different Jordan forms.

Step-by-step solution

Identify Eigenvalues of Matrix A

To understand if there exists an invertible matrix \( C \) we first need to identify the eigenvalues of matrix \( A = \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{bmatrix} \). Here, \( A \) is an upper triangular matrix, so its eigenvalues are the entries on its diagonal: 1, 1, and 1.

Identify Eigenvalues of Matrix B

The next step is to identify the eigenvalues of matrix \( B = \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). Matrix \( B \) is also upper triangular, hence its eigenvalues are: 1, 1, and 1.

Compare Eigenvalues of A and B

Both matrices \( A \) and \( B \) have identical eigenvalues (1, 1, and 1). However, to verify the existence of \( C \), we need to also consider eigenvectors and minimal polynomial.

Assess Jordan Form of A and B

Matrix \( A \) has a single Jordan block for the eigenvalue 1 of size 3 because it has a superdiagonal with all ones, indicating a single chain of generalized eigenvectors. \( B \), on the other hand, has two Jordan blocks of size 2 and 1 for the eigenvalue 1, indicating two chains. Matrices with different Jordan forms cannot be similar.

Conclusion on Invertible Matrix C

Since the Jordan forms of \( A \) and \( B \) are different, there cannot be any invertible matrix \( C \) such that \( C^{-1}AC = B \). They are not similar matrices.

Key concepts

Eigenvalues

Eigenvalues are crucial in understanding the behavior of matrices. For any given matrix, an eigenvalue is a number such that when it is subtracted from the diagonal elements of the matrix, and the determinant is zero, it provides meaningful insights into the properties of the matrix. Since matrix operations often involve Eigen decomposition, knowing how to find these values is useful. Consider the example of an upper triangular matrix, where eigenvalues can be directly taken from the diagonal elements, simplifying the task.For matrix \( A = \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{bmatrix} \), the eigenvalues are 1, 1, and 1. Similarly, the eigenvalues of matrix \( B = \begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \) are also 1, 1, and 1. The eigenvalues, though alike, might suggest similarity that is explored deeper in other properties.

Jordan Form

The Jordan form of a matrix is a canonical form that provides insights into the geometric and algebraic multiplicity of the eigenvalues. It is especially useful in understanding the structure of a matrix beyond just its eigenvalues.A Jordan block is characterized by having a single eigenvalue repeated on the diagonal, and ones on the superdiagonal. This structure informs us of the chains of generalized eigenvectors associated with each eigenvalue.For the matrix \( A \), all eigenvalues are 1, and we notice a single chain of generalized eigenvectors, forming a single Jordan block of size 3. However, for matrix \( B \), even though the eigenvalues are identical to those of \( A \), it has a different distribution, with a Jordan block of size 2 and another of size 1. This distinction reveals different matrix behaviors, key to understanding if two matrices can be similar.

Matrix Similarity

Matrix similarity involves checking whether two matrices are structurally identical in terms of their eigenvalues and eigenvectors. Two matrices are similar if there exists an invertible matrix \( C \) such that \( C^{-1}AC = B \), indicating that they represent the same linear transformation under different bases.Even when matrices \( A \) and \( B \) have the same eigenvalues, it doesn’t guarantee similarity. The crucial step involves comparing their Jordan forms. Matrix \( A \) and matrix \( B \) having different Jordan forms implies no invertible matrix \( C \) exists that can transform \( A \) into \( B \). Hence, the lack of similarity hinders any relation between the two, even with matching eigenvalues.

Invertible Matrix

An invertible matrix, often referred to as a non-singular matrix, is one that has an inverse, ensuring it can be classified as reversible. Mathematically, a matrix is invertible if there is another matrix that can multiply with it to produce the identity matrix.In our given problem, exploring the existence of an invertible matrix \( C \) which can link two matrices \( A \) and \( B \) via similarity, led us to realize such a matrix doesn’t exist due to differing Jordan forms. The inverse's role is central in achieving matrix transformations needed for recognizing similarity between matrices.The conclusion from our exploration is that without satisfying both identical eigenvalues and matching Jordan forms, an invertible matrix \( C \) cannot exist in this context.