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Chapter 3: Problem 34

If \(A\) is symmetric and invertible and \(A=L D L^{T}\) (with \(L\) unit lower triangular and \(D\) diagonal), prove that this factorization is unique. That is, prove that if we also have \(A=L_{1} D_{1} L_{1}^{T}\left(\text { with } L_{1} \text { unit lower triangular and } D_{1}\right.\) diagonal), then \(L=L_{1}\) and \(D=D_{1}.\)

Short Answer

Expert verified
The factorization is unique: \( L = L_1 \) and \( D = D_1 \).

Step by step solution

01

Analyze Factorization

We are given a symmetric and invertible matrix \( A \) with a factorization \( A = L D L^{T} \), where \( L \) is unit lower triangular and \( D \) is diagonal. We also have another factorization \( A = L_1 D_1 L_1^{T} \) with \( L_1 \) and \( D_1 \) having the same properties.
02

Express Equivalence of Factorizations

Since \( A = L D L^{T} = L_1 D_1 L_1^{T} \), we can equate the factorizations as \( L D L^{T} = L_1 D_1 L_1^{T} \).
03

Utilize Matrix Inversion

Since \( A \) is invertible, we infer that \( L \), \( D \), \( L_1 \), and \( D_1 \) must also support invertibility. Taking the inverse of both sides, we get \( (L_1^{-1})^T D_1^{-1} L_1^{-1} = (L^{-1})^T D^{-1} L^{-1} \).
04

Simplify and Contrast Components

Since \( D \) and \( D_1 \) are diagonal matrices, and \( L \) and \( L_1 \) are unit lower triangular, simplifying the equation leverages the uniqueness of a symmetric matrix factorization, particularly the properties of triangular matrices. Consider the constraint \( LD = L_1 D_1 \) obtained by matching terms.
05

Compare Triangular Matrix Entries

The uniqueness is assured by the fact that diagonal entries in \( LD \) and \( L_1 D_1 \) must match, such that the only form compatible with the constraints is \( L = L_1 \) and \( D = D_1 \). Non-diagonal entries confirm that other entries must align similarly due to the determinant properties not changing.
06

Conclude Uniqueness of Factorization

With the established conditions, and knowing the uniqueness of matrix inverses under the given properties, we conclude that the factorization is unique, which implies \( L = L_1 \) and \( D = D_1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Symmetric Matrix
A symmetric matrix is a square matrix that is identical to its transpose. That is, symmetric matrices satisfy the condition \(A = A^T\). One of the defining characteristics of symmetric matrices is this property of mirroring across the diagonal.
Symmetric matrices have several important properties:
  • All eigenvalues of a symmetric matrix are real, which means they can be plotted on the real number line.
  • Symmetric matrices are always diagonalizable. This property is crucial because diagonal matrices are much simpler to work with mathematically.
  • If a symmetric matrix is positive definite, it will have all positive eigenvalues.
These properties make symmetric matrices very useful in various applications such as physics, engineering, and computer graphics. In the context of matrix factorization, a symmetric matrix can often be expressed in terms of other matrices like lower triangular and diagonal matrices, as in the case of the Cholesky decomposition.
Triangular Matrix
Triangular matrices are matrices that have non-zero elements on either the upper or lower part of the matrix. A matrix is called lower triangular if all elements above the diagonal are zero, and it's called upper triangular if all elements below the diagonal are zero.
There are particular features of triangular matrices that make them highly valuable:
  • Easy computation of determinants: For a triangular matrix, the determinant is simply the product of its diagonal elements.
  • Solving linear equations: Systems represented by triangular matrices can be solved using forward or backward substitution, depending on whether the matrix is lower or upper triangular.
  • Inversion: The inverse of a triangular matrix is also triangular, which simplifies inversion processes.
In matrix factorizations like the ones described in textbook problems, triangular matrices play a significant role. For example, using a lower triangular matrix \(L\) to decompose other matrices is common in decompositions such as LDL decomposition, which is particularly useful when dealing with symmetric matrices.
Diagonal Matrix
A diagonal matrix is a matrix in which all entries outside the main diagonal are zero. The main diagonal can contain any numbers, including zeroes, which makes diagonal matrices much more straightforward than other types of matrices.
Key characteristics of diagonal matrices include:
  • Simplicity of matrix operations: Multiplying two diagonal matrices or finding the inverse of a diagonal matrix is much simpler compared to other matrices.
  • Eigenvalues and eigenvectors: The eigenvalues of a diagonal matrix are simply the diagonal entries themselves. This property makes working with diagonal matrices very straightforward in spectral theory.
  • Usefulness in decompositions: Diagonal matrices often feature in matrix decompositions, where simplifying computations is crucial, such as in singular value decomposition (SVD).
In a symmetrical and invertible matrix, factorized as \(A = L D L^{T}\), \(D\) is a diagonal matrix. It's important because it simplifies computations, helping demonstrate matrix properties like invertibility and uniqueness.
Invertible Matrix
An invertible matrix is a square matrix that has a corresponding matrix known as its inverse. When a matrix is multiplied by its inverse, the result is the identity matrix. This property is crucial for solving systems of equations and other mathematical problems.
Some vital points about invertible matrices include:
  • Determinant condition: A matrix is invertible if and only if its determinant is non-zero.
  • Unique solution: In linear algebra, systems of equations have a unique solution if the coefficient matrix is invertible.
  • Retaining properties: The inverse of a symmetric matrix is also symmetric. Similarly, inverses keep many properties of original matrices, such as the characteristic polynomials.
In matrix factorization, showing that a decomposed matrix (e.g., in the form \(LDL^{T}\)) is invertible implies specific and rigid structures. This property helps in proving the uniqueness of such factorizations, as seen in proving that \(L = L_1\) and \(D = D_1\) for the same symmetric invertible matrix \(A\).

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Most popular questions from this chapter

Prove that an \(m \times n\) matrix \(A\) has rank 1 if and only if \(A\) can be written as the outer product uv \(^{T}\) of a vector \(\mathbf{u}\) in \(\mathbb{R}^{m}\) and \(\mathbf{v}\) in \(\mathbb{R}^{n}\).

A population with four age classes has a Leslie matrix \(L=\left[\begin{array}{llll}0 & 1 & 2 & 5 \\ 0.5 & 0 & 0 & 0 \\ 0 & 0.7 & 0 & 0 \\ 0 & 0 & 0.3 & 0\end{array}\right] .\) If the initial population vector is \(\mathbf{x}_{0}=\left[\begin{array}{c}10 \\ 10 \\ 10 \\\ 10\end{array}\right],\) compute \(\mathbf{x}_{1}, \mathbf{x}_{2},\) and \(\mathbf{x}_{3}\).

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Draw a digraph that has the given adjacency matrix. $$\left[\begin{array}{lllll} 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \end{array}\right]$$

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