Question

Find the Cholesky decomposition of each of the following matrices. a. \(\left[\begin{array}{ll}4 & 3 \\ 3 & 5\end{array}\right]\) b. \(\left[\begin{array}{rr}2 & -1 \\ -1 & 1\end{array}\right]\) c. \(\left[\begin{array}{rrr}12 & 4 & 3 \\ 4 & 2 & -1 \\ 3 & -1 & 7\end{array}\right]\) d. \(\left[\begin{array}{rrr}20 & 4 & 5 \\ 4 & 2 & 3 \\ 5 & 3 & 5\end{array}\right]\)

Short answer

a: Possible, b: Not possible, c: Possible, d: Possible.

Step-by-step solution

Prepare the Matrix for Decomposition a

First, let's identify the matrix for part (a): \[\begin{bmatrix}4 & 3 \3 & 5\end{bmatrix}\]In Cholesky decomposition, our goal is to express this matrix as the product of a lower triangular matrix and its transpose.

Compute the Cholesky Decomposition for a

The Cholesky decomposition of a matrix \( A \) is represented as \( A = LL^T \), where \( L \) is a lower triangular matrix. For this 2x2 matrix, assume:\[L = \begin{bmatrix}l_{11} & 0 \l_{21} & l_{22}\end{bmatrix}\]Calculate:- \( l_{11} = \sqrt{4} = 2 \)- \( l_{21} = 3 / 2 = 1.5 \)- \( l_{22} = \sqrt{5 - 1.5^2} = \sqrt{2.75} = 1.6583 \)Therefore, the Cholesky factor \( L \) is:\[\begin{bmatrix}2 & 0 \1.5 & 1.6583\end{bmatrix}\]

Prepare the Matrix for Decomposition b

Next, consider the matrix for part (b): \[\begin{bmatrix}2 & -1 \-1 & 1\end{bmatrix}\]

Check for Positive Definiteness of b

Before proceeding with decomposition, ensure the matrix is positive definite by checking if all leading principal minors are positive. This matrix is not positive definite since the determinant \( (2 \cdot 1) - (-1)^2 = 1 \) is non-positive.

Cholesky Decomposition Not Possible for b

Since the matrix \( \begin{bmatrix} 2 & -1 \ -1 & 1 \end{bmatrix} \) is not positive definite, Cholesky decomposition is not possible.

Prepare the Matrix for Decomposition c

Consider the next matrix:\[\begin{bmatrix}12 & 4 & 3 \4 & 2 & -1 \3 & -1 & 7\end{bmatrix}\]

Compute the Cholesky Factors for c

Assume that \( L \) has lower triangular form:\[L = \begin{bmatrix}l_{11} & 0 & 0 \l_{21} & l_{22} & 0 \l_{31} & l_{32} & l_{33}\end{bmatrix}\]Calculate:\( l_{11} = \sqrt{12} = 3.4641 \),\( l_{21} = 4 / l_{11} = 1.1547 \),\( l_{31} = 3 / l_{11} = 0.8660 \),\( l_{22} = \sqrt{2 - (1.1547)^2} = 1 \),\( l_{32} = (3\cdot -1) - (0.8660\cdot 1) = -2 \),\( l_{33} = \sqrt{7 - 0.8660^2 - (-2)^2} = 2.6458 \).The result is:\[\begin{bmatrix}3.4641 & 0 & 0 \1.1547 & 1 & 0 \0.8660 & -2 & 2.6458\end{bmatrix}\]

Prepare the Matrix for Decomposition d

Finally, consider the matrix:\[\begin{bmatrix}20 & 4 & 5 \4 & 2 & 3 \5 & 3 & 5\end{bmatrix}\]

Compute the Cholesky Factors for d

Use a similar lower triangular form:\[L = \begin{bmatrix}l_{11} & 0 & 0 \l_{21} & l_{22} & 0 \l_{31} & l_{32} & l_{33}\end{bmatrix}\]Calculate:\( l_{11} = \sqrt{20} = 4.4721 \),\( l_{21} = 4 / 4.4721 = 0.8944 \),\( l_{31} = 5 / 4.4721 = 1.1180 \),\( l_{22} = \sqrt{2 - 0.8944^2} = 0.4472 \),\( l_{32} = (5\cdot 3) - (1.1180\cdot 0.4472) = 2.2361 \),\( l_{33} = \sqrt{5 - 1.1180^2 - 2.2361^2} = 1 \).Hence:\[\begin{bmatrix}4.4721 & 0 & 0 \0.8944 & 0.4472 & 0 \1.1180 & 2.2361 & 1\end{bmatrix}\]

Key concepts

Positive Definite Matrix

A Positive Definite Matrix is a key concept in linear algebra, particularly when dealing with matrix decompositions like Cholesky Decomposition. A square matrix is said to be positive definite if for any non-zero column vector \(x\), the scalar \(x^T A x\) is always positive, where \(x^T\) is the transpose of \(x\) and \(A\) is the matrix in question. This means the matrix has properties like all its eigenvalues being positive.

• Positive definiteness is necessary for Cholesky Decomposition.
• This property ensures the existence of a unique triangular matrix when factoring.
• For example, if a matrix's leading principal minors are all positive, it suggests positive definiteness.

In our original exercise, matrix (b) failed to meet this requirement, as its determinant was non-positive, making Cholesky decomposition infeasible.

Lower Triangular Matrix

A Lower Triangular Matrix is a type of matrix that plays a pivotal role in the computation of Cholesky Decomposition. It is structured so that all elements above the diagonal are zero, leaving non-zero elements only on and below the diagonal.

• The diagonal elements and below can be any number, while the rest above must be zero.
• These matrices are easy to work with, especially for solving systems of linear equations.
• In Cholesky decomposition, the original matrix is expressed as \(A = LL^T\), where \(L\) is a lower triangular matrix.

Being lower triangular makes computations efficient and is essential for ensuring numerical stability in algorithms. This form simplifies operations like forward substitution.

Matrix Factorization

Matrix Factorization involves breaking down a matrix into simpler, component matrices. This is essential in solving linear equations, simplifying computations, and computational optimizations.

• Cholesky Decomposition is one form of matrix factorization, suitable for positive definite matrices.
• It splits a matrix into a product of a lower triangular matrix and its transpose.
• This factorization is preferred for its efficiency with numerical methods.

In the exercise, we factored matrices into lower triangular matrices, illustrating this concept with examples that show both success and failure due to positive definiteness requirements.

Numerical Linear Algebra

Numerical Linear Algebra is the branch of mathematics that focuses on numerical solutions to linear algebra problems. It plays a crucial role in scientific computing and data science.

• This field involves algorithms for performing matrix operations to high precision.
• Cholesky Decomposition is an effective algorithm used in numerical methods, especially with positive definite matrices.
• Application areas are diverse, including solving equations in physics, engineering, and even machine learning.

By using techniques from numerical linear algebra, we ensure that our matrix operations, like the ones in the original exercise, are performed efficiently and accurately, even for large-scale computations.