Question

We know that similar matrices have the same eigenvalues (in fact, they have the same characteristic polynomial). There are many examples that show the converse is not true; that is, there are examples of matrices \(A\) and \(B\) that have the same characteristic polynomial but are not similar. Show that the following matrices \(A\) and \(B\) cannot be similar: $$ A=\left[\begin{array}{ll} 1 & 0 \\ 3 & 1 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$

Short answer

Based on the given matrices \(A\) and \(B\), the matrices have the same characteristic polynomial with a single eigenvalue \(\lambda = 1\). However, matrix \(A\) is not diagonalizable, and matrix \(B\) is already a diagonal matrix. This proves that the matrices are not similar, even though they share the same characteristic polynomial.

Step-by-step solution

Calculate the trace of both matrices

To compute the trace of a matrix, we simply sum up its diagonal elements. So for the given matrices, we have: $$ \text{Trace}(A) = 1 + 1 = 2 \\ \text{Trace}(B) = 1 + 1 = 2 $$ Since the traces of both matrices are equal, we need to check their determinants.

Calculate the determinant of both matrices

To compute the determinant of a 2x2 matrix, we use the formula: \(\text{det}(M) = M_{11}M_{22} - M_{12}M_{21}\). Applying this formula to the given matrices, we obtain: $$ \text{det}(A) = (1)(1) - (0)(3) = 1 \\ \text{det}(B) = (1)(1) - (0)(0) = 1 $$ The determinants of the matrices are also equal. Since the characteristic polynomial is determined by the trace and determinant, the matrices \(A\) and \(B\) have the same characteristic polynomial. Although in general this is not a sufficient condition for similarity, in this specific case it can also be used to show that these specific matrices \(A\) and \(B\) are not similar.

Check if they are diagonalizable

Compute the eigenvalues for both matrices by solving the characteristic equation \(\text{det}(A - \lambda I) = 0\) and \(\text{det}(B - \lambda I) = 0\). For both matrices \(A\) and \(B\), this equation is: $$ (1 - \lambda)(1 - \lambda) - (0)(0) = 0 $$ Solving the equation, we see that both matrices have a single eigenvalue, \(\lambda = 1\). Since \(A\) is a 2x2 matrix with only one eigenvalue, it is not diagonalizable. However, the matrix \(B\) is already a diagonal matrix. Therefore, \(A\) and \(B\) do not share the same diagonal form and so they cannot be similar.

Conclusion

In this specific case, even though the given matrices \(A\) and \(B\) have the same characteristic polynomial, we have shown that they are not similar by observing that matrix \(A\) is not diagonalizable while matrix \(B\) is already a diagonal matrix.

Key concepts

Matrices

Matrices are fundamental concepts in linear algebra. They are rectangular arrays of numbers arranged in rows and columns. In our example, both matrices have 2 rows and 2 columns, also known as 2x2 matrices. Matrices are used to perform a variety of operations in mathematics, including rotations, translations, and scaling transformations. They can also represent systems of linear equations. Key features of matrices include:

• Their size or dimension, which is given by the number of rows and columns.
• Operations such as matrix addition, multiplication, and finding inverses.
• Special types of matrices, like diagonal matrices, identity matrices, and triangular matrices.

Mathematics involving matrices becomes useful when solving real-world problems that can be modeled by linear transformations, enabling efficient data manipulation and transformation.

Eigenvalues

Eigenvalues are vital in understanding many properties of matrices. An eigenvalue of a matrix is a scalar that indicates how much the corresponding eigenvector is stretched during a linear transformation represented by the matrix. In our example, both matrices have eigenvalues that are solutions of the equation \text{det}(A - \lambda I) = 0\. To find these eigenvalues, we solve:\((1 - \lambda)(1 - \lambda) = 0\), leading to a single eigenvalue \(\lambda = 1\).Understanding eigenvalues helps in:

• Determining the stability of a system.
• Identifying the inherent 'directions' in which transformations act.
• Facilitating diagonalization, where possible.

Eigenvalues play a critical role in various applications, including physics, engineering, and computer science, where they help simplify complex transformations.

Characteristic Polynomial

The characteristic polynomial of a matrix is a special polynomial, derived from the determinant of the matrix minus a scalar multiple of the identity matrix. This polynomial provides insight into the eigenvalues of the matrix. For matrix \(A\), the characteristic polynomial is computed as follows: \(\text{CharacterPoly}(A) = \text{det}(A - \lambda I)\).In our specific example, \(\text{det}\begin{vmatrix}1-\lambda & 0 \3 & 1-\lambda\end{vmatrix} = (1-\lambda)^2\).This polynomial results in solutions that are the eigenvalues of the matrix. The importance of the characteristic polynomial includes:

• Providing eigenvalues through its roots.
• Helping determine if a matrix is diagonalizable.
• Indicating similarity between matrices, though similar matrices need more than having identical characteristic polynomials.

Understanding the characteristic polynomial aids in the deeper study of linear transformations and their effects.

Diagonalization

Diagonalization is a powerful method used in linear algebra to simplify matrices. It involves expressing a matrix in terms of a diagonal matrix, potentially making computations easier. A matrix is diagonalizable if it can be represented as \(PDP^{-1}\), where \(D\) is a diagonal matrix and \(P\) is an invertible matrix. In our scenario, we checked whether matrix \(A\) could be diagonalized. However, since it only has one eigenvalue, \(\lambda = 1\), and isn't a diagonal matrix already, it doesn't meet the requirements for diagonalization. Conversely, matrix \(B\) is already diagonal and thus diagonalizable by itself. Diagonalization is valuable for:

• Simplifying matrix raised to powers.
• Facilitating the computation of matrix functions.
• Improving understanding of the transformation represented by a matrix.

When a matrix can be diagonalized, it simplifies the computation process and improves the interpretation of linear transformations.