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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 \(A=L_{1} D_{1} L_{1}^{T}\) (with \(L_{1}\) unit lower triangular and \(D_{1}\) 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

Express the two factorizations

We are given two different factorizations of the matrix \(A\): \(A = LDL^T\) and \(A = L_1D_1L_1^T\). Our goal is to show that \(L = L_1\) and \(D = D_1\). Both factorizations represent the same matrix \(A\).
02

Equating the factorizations

Since both expressions equal \(A\), we have \(LDL^T = L_1D_1L_1^T\). Equating these two, involves matching both the lower triangular components and the diagonal components.
03

Analyze the inverse expressions

Since \(A\) is invertible, both \(LDL^T\) and \(L_1D_1L_1^T\) must be invertible. Thus, their inverses also exist, implying \(D\) and \(D_1\) are invertible diagonal matrices.
04

Use the property of triangular matrices

Given that \(L\) and \(L_1\) are unit lower triangular matrices, any left multiplication with their inverses (also unit lower triangular) retains the unit lower triangular structure.
05

Equate corresponding elements

By focusing on the diagonal and the lower triangular structure, since \(LD = L_1D_1(L_1^{'})L^T\) and both \(L\) and \(L_1\) are unit lower triangular, each corresponding lower triangular component and diagonal must be identical. It follows that \(L = L_1\) and \(D = D_1\).
06

Conclusion

Given the properties of unit lower triangular matrices and invertible diagonal matrices, the unique representation of the factorization is established as \(L = L_1\) 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 special type of square matrix that is equal to its transpose. This means for any matrix \( A \), \( A = A^T \). In simpler terms, the elements across the main diagonal mirror each other. For instance, if the entry in the first row, second column is 3, then the entry in the second row, first column must also be 3.
  • Key Property: The concept of symmetry means \( A[i, j] = A[j, i] \) for all elements.
  • Applications: They occur frequently in areas such as physics, optimization, and statistics.
  • Characteristic: Symmetric matrices have real eigenvalues, which is important in many applications.
Understanding symmetric matrices is crucial because they often simplify complex problems and provide stable solutions.
Invertible Matrix
An invertible matrix, also known as a non-singular or non-degenerate matrix, is one that has an inverse. This means there is another matrix, commonly denoted as \(A^{-1}\), such that when multiplied with the original matrix \(A\), it results in the identity matrix \(I\). Mathematically, \(AA^{-1} = A^{-1}A = I\).
  • Condition: A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible.
  • Significance: Invertible matrices allow for solving systems of linear equations, as they can reverse or "undo" transformations.
  • Applications: These matrices are crucial in linear algebra applications, including engineering and computer graphics.
The concept of matrix invertibility is fundamental in mathematics, as it ensures processes can be reversed, providing flexibility in manipulation and analysis of equations.
Lower Triangular Matrix
A lower triangular matrix is a type of square matrix where all elements above the main diagonal are zero. This means non-zero elements can only appear on or below the main diagonal.
  • Structure: If \(A\) is a lower triangular matrix, then \(A[i, j] = 0\) for all \(i < j\).
  • Unit Lower Triangular: A unit lower triangular matrix is a special case where all the diagonal elements are 1.
  • Properties: Multiplying two lower triangular matrices results in another lower triangular matrix.
This structure is valuable because it simplifies calculations, particularly in solving systems of equations where back substitution can efficiently be used.
Diagonal Matrix
A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. Only the elements on the main diagonal can be non-zero.
  • Form: A diagonal matrix \(D\) has the property that \(D[i, j] = 0\) for \(i eq j\).
  • Invertibility: A diagonal matrix is invertible if and only if all its diagonal elements are non-zero.
  • Operations: Diagonal matrices are straightforward to manipulate because operations like multiplication and finding powers are simplified to dealing with the diagonal elements only.
These matrices are particularly significant as they represent scaling transformations and are crucial in matrix factorization methods, where they simplify the mathematical expressions and computations.

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Most popular questions from this chapter

Find a \(P^{T} L U\) factorization of the given matrix A. $$A=\left[\begin{array}{rrrr}0 & -1 & 1 & 3 \\\\-1 & 1 & 1 & 2 \\\0 & 1 & -1 & 1 \\\0 & 0 & 1 & 1\end{array}\right]$$

Suppose the Leslie matrix for the VW beetle is \(L=\left[\begin{array}{lll}0 & 0 & 20 \\ s & 0 & 0 \\ 0 & 0.5 & 0\end{array}\right] .\) Investigate the effect of varying the survival probability \(s\) of the young beetles.

Answer by considering the matrix with the given vectors as its columns. Do \(\left[\begin{array}{r}1 \\ -1 \\\ 3\end{array}\right],\left[\begin{array}{r}-1 \\ 5 \\\ 1\end{array}\right],\left[\begin{array}{r}1 \\ -3 \\ 1\end{array}\right]\) form a basis for \(\mathbb{R}^{3} ?\)

Find the standard matrix of the given linear transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\) Reflection in the line \(y=x\)

Prove that every unit lower triangular matrix is invertible and that its inverse is also unit lower triangular.
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