Question

Let \(A=\left[\begin{array}{cc}1 & a \\ a & b\end{array}\right]\). If \(a^{2}

Short answer

Matrix \(A\) is positive definite and its Cholesky factorization is \(L = \begin{bmatrix} 1 & 0 \\ a & \sqrt{b-a^2} \end{bmatrix}\).

Step-by-step solution

Positive Definite Definition

To prove that matrix \(A\) is positive definite, we need to show that for any non-zero vector \(x = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\), \(x^T A x > 0\). Based on the structure of \(A\), the expression \(x^T A x\) will evaluate to:\(x^T A x = x_1^2 + 2ax_1x_2 + bx_2^2.\)

Apply the Condition

Given the inequality \(a^2 < b\), rewrite \(x^T A x\) as a quadratic in \(x_2\):\[ x^T A x = (x_1 + ax_2)^2 + (b-a^2)x_2^2. \]Notice that the first term is always non-negative as it is a square, and the second term is positive for any non-zero \(x_2\). Thus, \(x^T A x > 0\) for all non-zero \(x\), confirming \(A\) is positive definite.

Setup for Cholesky Factorization

We aim to express \(A = LL^T\), where \(L\) is a lower triangular matrix of the form \[L = \begin{bmatrix} l_{11} & 0 \ l_{21} & l_{22} \end{bmatrix}.\] Equating this to matrix \(A\), we have: \[ \begin{bmatrix} l_{11}^2 & l_{11} l_{21} \ l_{11} l_{21} & l_{21}^2 + l_{22}^2 \end{bmatrix} = \begin{bmatrix} 1 & a \ a & b \end{bmatrix}. \]

Solve for Elements of L

From the first element, solve \(l_{11}^2 = 1\), giving \(l_{11} = 1\). From the second element, solve \(l_{11} l_{21} = a\), yielding \(l_{21} = a\). From the last equation, \(l_{21}^2 + l_{22}^2 = b\), substitute \(l_{21} = a\) to get \(a^2 + l_{22}^2 = b\), which results in \(l_{22} = \sqrt{b - a^2}\).

Verify the Cholesky Factorization

We now have \(L = \begin{bmatrix} 1 & 0 \ a & \sqrt{b-a^2} \end{bmatrix}\). Calculate \(LL^T\) to verify:\[LL^T = \begin{bmatrix} 1 & 0 \ a & \sqrt{b-a^2} \end{bmatrix} \begin{bmatrix} 1 & a \ 0 & \sqrt{b-a^2} \end{bmatrix} = \begin{bmatrix} 1 & a \ a & b \end{bmatrix} = A,\]confirming the correctness of the factorization.

Key concepts

Positive Definite Matrices

Positive definite matrices are an important concept in linear algebra and are central to understanding Cholesky Decomposition. For a symmetric matrix to be positive definite, it must satisfy a specific condition: for any non-zero vector \( x \), the quadratic form \( x^T A x \) is always greater than zero. This ensures that the matrix can represent a system of equations that have unique solutions, which is crucial in numerical methods.In the exercise, the matrix \( A = \begin{bmatrix} 1 & a \ a & b \end{bmatrix} \) was tested for positive definiteness by examining the quadratic form \( x^T A x = x_1^2 + 2ax_1x_2 + bx_2^2 \). By factorizing this expression into \( (x_1 + ax_2)^2 + (b-a^2)x_2^2 \) and using the condition \( a^2 < b \), we established it is positive definite because both terms in the expression are non-negative or explicitly positive. The ability to confirm positive definiteness allows us to use algorithms like Cholesky Decomposition, ensuring numerical stability.

Matrix Factorization

Matrix factorization is a pivotal technique in linear algebra, breaking down a complex matrix into simpler components, which simplifies many matrix computations. The Cholesky Decomposition is one such method, specifically applied to positive definite matrices, where a symmetric matrix \( A \) is decomposed into a product \( LL^T \), where \( L \) is a lower triangular matrix.For the given matrix \( A = \begin{bmatrix} 1 & a \ a & b \end{bmatrix} \), we set \( L \) as \( \begin{bmatrix} l_{11} & 0 \ l_{21} & l_{22} \end{bmatrix} \) and derived the values step by step:

• \( l_{11}^2 = 1 \), leading to \( l_{11} = 1 \).
• \( l_{11} l_{21} = a \), leading to \( l_{21} = a \).
• \( l_{21}^2 + l_{22}^2 = b \), giving \( l_{22} = \sqrt{b-a^2} \).

Thus, \( L \) was found to be \( \begin{bmatrix} 1 & 0 \ a & \sqrt{b-a^2} \end{bmatrix} \), and multiplying \( L \) by its transpose \( L^T \) gets back our original matrix \( A \), showcasing the effectiveness of Cholesky factorization.

Quadratic Forms

Quadratic forms are expressions that involve variables raised to the second power, often seen in the context of matrices as \( x^T A x \), where \( x \) is a vector and \( A \) is a matrix. Quadratic forms reveal much about the characteristics of matrices, such as definiteness (whether a matrix is positive definite).The exercise used the quadratic form \( x^T A x = x_1^2 + 2ax_1x_2 + bx_2^2 \) to examine the positive definiteness of \( A \). By rewriting it as \( (x_1 + ax_2)^2 + (b-a^2)x_2^2 \), we demonstrated that the matrix is positive definite if \( a^2 < b \). This transformation also facilitates other matrix operations, such as computing the Cholesky factorization, by revealing how different matrix components interact.Understanding quadratic forms is crucial because they appear in various applications, including optimization problems and stability analyses, representing complex relationships in a simplified mathematical form.