Question

Assume the \(2 \times 2\) matrix \(A\) is similar to an upper triangular matrix. If \(\operatorname{tr} A=0=\operatorname{tr} A^{2},\) show that \(A^{2}=0\)

Short answer

Since \( \operatorname{tr} A = 0 \) and \( \operatorname{tr} A^2 = 0 \), \( A^2 = 0 \) by Cayley-Hamilton theorem.

Step-by-step solution

Understanding Matrix Similarity

If matrix \( A \) is similar to an upper triangular matrix \( U \), it means there exists an invertible matrix \( P \) such that \( A = PUP^{-1} \). Also, the eigenvalues of \( A \) are on the diagonal of \( U \).

Analyzing Given Conditions

We know \( \operatorname{tr} A = 0 \), which implies that the sum of the eigenvalues (say \( \lambda_1 \) and \( \lambda_2 \)) of \( A \) is zero: \( \lambda_1 + \lambda_2 = 0 \). Additionally, we know \( \operatorname{tr} A^2 = 0 \). This implies that \( \lambda_1^2 + \lambda_2^2 = 0 \).

Determining Eigenvalues

Given that \( \lambda_1 + \lambda_2 = 0 \), we have \( \lambda_2 = -\lambda_1 \). Substituting in the second trace condition \( \lambda_1^2 + \lambda_2^2 = 0 \), we get \( \lambda_1^2 + (-\lambda_1)^2 = 0 \). Simplifying leads to \( 2\lambda_1^2 = 0 \), so \( \lambda_1^2 = 0 \), and therefore \( \lambda_1 = 0 \). Hence, \( \lambda_2 = 0 \) as well.

Finding Matrix Polynomial

The characteristic polynomial of \( A \), given the eigenvalues, is \( (x - \lambda_1)(x - \lambda_2) = x^2 \). Since both eigenvalues are zero, this polynomial is simply \( x^2 \).

Using Cayley-Hamilton Theorem

According to the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation. For matrix \( A \), this implies \( A^2 = 0 \).

Key concepts

Eigenvalues

Eigenvalues are significant in understanding many properties of matrices. For any square matrix, the eigenvalues are the special set of scalars such that when subtracted from the diagonal entries of the matrix, the determinant becomes zero. In simpler terms, they are values that, when plugged into the matrix equation, do not change direction within the space.

• Finding eigenvalues involves solving the equation \( \det(A - \lambda I) = 0 \)
• In this problem, the matrix \( A \) is similar to an upper triangular matrix, meaning its eigenvalues appear on the diagonal.

Given the conditions \( \operatorname{tr} A = 0 \) and \( \operatorname{tr} A^2 = 0 \), we derived that the sum and squares of the eigenvalues are zero.
For a 2x2 matrix such as this, knowing \( \lambda_1 + \lambda_2 = 0 \) and \( \lambda_1^2 + \lambda_2^2 = 0 \) mathematically leads us to determine that both eigenvalues are zero, \( \lambda_1 = \lambda_2 = 0 \).
This forms the foundation by which we verify or apply the steps of the Cayley-Hamilton Theorem.

Cayley-Hamilton Theorem

The Cayley-Hamilton Theorem is a foundational result in linear algebra that states every square matrix satisfies its own characteristic equation.
For example, given a square matrix \( A \) and its characteristic polynomial, \( p(x) \), according to the Cayley-Hamilton theorem, \( A \) itself will satisfy \( p(A) = 0 \).

• In simpler language, substitute the matrix \( A \) for \( x \) in its characteristic polynomial and the result will be the zero matrix.
• This holds true for matrices of any size, including our case here with a 2x2 matrix.

In the problem, we identified that the characteristic polynomial of \( A \) is \( x^2 \) because both eigenvalues are zero.
Thus, applying the Cayley-Hamilton Theorem, the equation transforms into \( A^2 = 0 \), revealing that the next powers of \( A \) also turn into zero matrices,
demonstrating how the theorem helps elucidate properties of matrices via their polynomials.

Characteristic Polynomial

The characteristic polynomial is a polynomial which is derived from a square matrix and is crucial in identifying the eigenvalues of that matrix.
For a given matrix \( A \), the characteristic polynomial \( p(x) \) is determined by \( \det(A - xI) \) where \( I \) is the identity matrix of the same dimension.

• This polynomial provides the eigenvalues of \( A \) as roots.
• In our example, having \( \lambda_1 = 0 \) and \( \lambda_2 = 0 \), the characteristic polynomial becomes \( x^2 \).

The simple nature of this polynomial in this particular problem indicates the redundancy of power matrices higher than itself.
The polynomial generates vital knowledge about the dynamics and properties inherent to \( A \), especially with the aid of the Cayley-Hamilton Theorem,
which utilizes the polynomial to directly inform us about \( A^2 = 0 \). The simplicity of the characteristic polynomial is central to proving certain conditions like \( A^2 = 0 \) seen in this exercise.