Question

If \(A\) is positive definite, show that \(A=C C^{T}\) where \(C\) has orthogonal columns.

Short answer

A positive definite matrix can be decomposed into an orthogonal column product using Cholesky decomposition.

Step-by-step solution

Understanding Positive Definite Matrices

A symmetric matrix \(A\) is positive definite if it satisfies \(x^T Ax > 0\) for all non-zero vectors \(x\). This means \(A\) can be decomposed into a product of matrices such that each value is greater than zero, confirming non-negative eigenspaces.

Cholesky Decomposition Concept

The Cholesky decomposition is used to express a symmetric positive definite matrix \(A\) as \(A = LL^T\) where \(L\) is a lower triangular matrix with positive diagonal entries. This is a key tool to represent \(A\) in forms involving simpler components.

Change of Basis Interpretation

By applying the Cholesky decomposition approach, consider the matrix \(C\) defined such that \(C = LQ\), where \(Q\) is an orthogonal matrix. Here, \(Q\) satisfies \(Q^TQ = I\) (the identity matrix), which maintains orthogonality in the columns of \(C\).

Forming and Adjusting \(C\)

Reshape \(C\) by ensuring its columns are made of orthogonal vectors, as per \(C = LQ\). The product \(C\) constructs \(A\) into \(A = (LQ)(LQ)^T\), essentially \(A = C C^T\) with orthogonally distributed outputs from \(Q\).

Validation and Conclusion

Double-check \(LL^T\) by reconstructing with orthogonal scaled basis to ensure \(A = C C^T\) holds true with \(C\) bearing orthogonal columns, hence showing correctness through the decomposition method applied specifically.

Key concepts

Positive Definite Matrices

A positive definite matrix is a special type of square matrix. The defining property is that it always results in a positive number when you multiply a vector by it in a certain way. Specifically, for a symmetric matrix \(A\), it is positive definite if \(x^T Ax > 0\) for any non-zero vector \(x\).
This property is quite important because it ensures that the matrix behaves well with respect to the vectors it interacts with.
Some key characteristics of positive definite matrices include:

• The matrix must be symmetric. This means it looks the same across its diagonal.
• All its eigenvalues are positive. Eigenvalues are special numbers associated with a matrix that give us a lot of information about its behavior.
• They are intimately related to real-world phenomena where stability and positivity are required.

Understanding these characteristics can help you identify positive definite matrices easily, facilitating matrix operations such as the Cholesky Decomposition.

Orthogonal Matrices

Orthogonal matrices are essential in linear algebra because they preserve the length of vectors during transformation. A matrix \(Q\) is orthogonal if its transpose is also its inverse, indicated by \(Q^TQ = I\), where \(I\) is the identity matrix.
This property means that if you apply an orthogonal matrix to a vector, you don't change its length or magnitude, only its direction.
Key properties include:

• Columns (or rows) of an orthogonal matrix are orthonormal vectors. Orthonormal vectors are both orthogonal (perpendicular) and normalized (having length 1).
• The determinant of an orthogonal matrix is either +1 or -1. This indicates that applying the matrix involves either rotation or reflection.
• They are widely used in numerical computations to ensure numerical stability.

Orthogonal matrices simplify computations like decompositions, which come into play in solving systems of equations and optimization problems, making them a versatile tool in mathematics and engineering.

Matrix Decomposition

Matrix decomposition involves breaking down a matrix into simpler components. This technique is fundamental in solving complex matrix equations efficiently. One of the most renowned methods of matrix decomposition is the Cholesky Decomposition.
Cholesky Decomposition specifically applies to symmetric positive definite matrices and expresses a matrix \(A\) as \(A = LL^T\), where \(L\) is a lower triangular matrix with positive diagonal elements.
The decomposition method allows:

• Efficient numerical solutions to linear systems. Solving \(LL^Tx = b\) can be broken down easily into two systems with triangular matrices.
• Enhanced computational speed and accuracy, especially in algorithms for machine learning and statistics.
• Simplifications in various mathematical operations, like eigenvalue computations and matrix inversions.

By using matrix decomposition, complex matrices become manageable, enabling numerous practical applications in science and technology, demonstrating the power of manipulating matrices into more tractable forms.