/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Let \(A\) be a positive definite... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 30

Let \(A\) be a positive definite symmetric matrix. Show that there exists an invertible matrix \(B\) such that \(A=\) \(B^{T} B .[\text {Hint: Use the Spectral Theorem to write } A=\) \(Q D Q^{T} .\) Then show that \(D\) can be factored as \(C^{T} C\) for some invertible matrix \(C .]\)

Short Answer

Expert verified
Matrix \(A\) can be expressed as \(B^T B\) using an invertible matrix \(B\).

Step by step solution

01

Use the Spectral Theorem

The Spectral Theorem states that any symmetric matrix \(A\) can be factored as \(A = QDQ^T\), where \(Q\) is an orthogonal matrix (\(Q^T = Q^{-1}\)) and \(D\) is a diagonal matrix with the eigenvalues of \(A\) on its diagonal.
02

Recognize Properties of Positive Definite Matrices

Since \(A\) is a positive definite matrix, all its eigenvalues are positive. This means the diagonal elements of \(D\), which are the eigenvalues of \(A\), are positive.
03

Factorize D as \(C^T C\)

Since \(D\) is a diagonal matrix with positive entries, it can be factorized as \(D = C^T C\), where \(C\) is a diagonal matrix with the positive square roots of the eigenvalues of \(A\) on its diagonal. Therefore, for each diagonal element \(d_i\) of \(D\), we have \(c_i = \sqrt{d_i}\).
04

Express \(B\) in Terms of \(Q\) and \(C\)

Now, define the matrix \(B = QC\). This matrix \(B\) is invertible since both \(Q\) and \(C\) are invertible. Then, \(B^T = C^T Q^T\).
05

Verify \(A = B^T B\)

Calculate \(B^T B = C^T Q^T QC = C^T C = D\). Thus, \(A = QDQ^T = (QC)^T (QC) = B^T B\), showing that such a matrix \(B\) exists.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Spectral Theorem
The Spectral Theorem is a pivotal concept in linear algebra, especially when dealing with symmetric matrices. It states that any symmetric matrix can be expressed in the form \(A = QDQ^T\), where:
  • \(A\) is the symmetric matrix in question.
  • \(Q\) is an orthogonal matrix, meaning \(Q^T = Q^{-1}\).
  • \(D\) is a diagonal matrix containing the eigenvalues of \(A\) along its diagonal.
This theorem is crucial because it simplifies many problems in linear algebra by allowing us to express matrices in terms of simpler components. These components, the orthogonal matrix \(Q\) and diagonal matrix \(D\), make it easier to understand and manipulate the properties of \(A\). Because of the nature of an orthogonal matrix, this factorization keeps the geometric properties of the matrix intact.
symmetric matrix
A symmetric matrix is a square matrix that is equal to its transpose. In other words, for a matrix \(A\), it is symmetric if \(A = A^T\). This property results in several interesting features:
  • All eigenvalues of a symmetric matrix are real numbers.
  • Symmetric matrices are diagonalizable using an orthogonal matrix, as indicated by the Spectral Theorem.
  • The symmetry of a matrix also implies that it has a set of orthogonal eigenvectors.
These characteristics make symmetric matrices very useful in various applications, from solving systems of equations to optimizing certain functions. They often represent steady and predictable systems, which is why they appear frequently in physical applications and statistical analyses.
orthogonal matrix
An orthogonal matrix is a type of square matrix where the columns and rows are orthogonal unit vectors. Formally, a matrix \(Q\) is orthogonal if it satisfies the condition \(Q^TQ = QQ^T = I\), where \(I\) is the identity matrix. Key properties of orthogonal matrices include:
  • They preserve both the length and the angle of vectors upon transformation, hence they represent rotations and reflections.
  • The inverse of an orthogonal matrix is simply its transpose, \(Q^{-1} = Q^T\).
  • They are involved in the diagonalization of symmetric matrices, which serves as a basis for the Spectral Theorem.
These attributes mean orthogonal matrices are incredibly useful when working with transformations and are central to solving many linear algebra problems effectively.
eigenvalues
Eigenvalues are scalar values related to a matrix that provide significant insight into the matrix's properties. Mathematically, a number \(\lambda\) is considered an eigenvalue of a matrix \(A\) if there exists a non-zero vector \(v\) such that \(Av = \lambda v\). In simpler terms:
  • Eigenvalues represent the factors by which the eigenvectors are scaled during a linear transformation.
  • For symmetric matrices, all eigenvalues are real, and they can be arranged along the diagonal of a diagonal matrix as highlighted in the Spectral Theorem.
  • For positive definite matrices, as in the original exercise, all eigenvalues are not just real but also positive.
  • The positivity ensures that certain factorizations, such as Cholesky decomposition, are possible.
Understanding eigenvalues is essential in diverse fields, from stabilizing solutions in differential equations to determining the principal components in data analysis.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the orthogonal projection of v onto the subspace \(W\) spanned by the vectors \(\mathbf{u}_{\mathbf{r}}\) (You may assume that the vectors \(\mathbf{u}_{i}\) are orthogonal. ) $$\mathbf{v}=\left[\begin{array}{r} 7 \\ -4 \end{array}\right], \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right]$$

Diagonalize the quadratic forms by finding an orthogonal matrix \(Q\) such that the change of variable \(\mathbf{x}=\) Qy transforms the given form into one with no cross-product terms. Give Q and the new quadratic form. $$7 x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+8 x_{1} x_{2}+8 x_{1} x_{3}-16 x_{2} x_{3}$$

Find the orthogonal decomposition of v with respect to \(W\). $$\mathbf{v}=\left[\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right], W=\operatorname{span}\left(\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\right)$$

Let \(W\) be a subspace of \(\mathbb{R}^{n},\) and let \(\mathbf{x}\) be a vector in \(\mathbb{R}^{n}\). Prove that \(x\) is orthogonal to \(W\) if and only if \(\operatorname{proj}_{w}(\mathbf{x})=0\).

Diagonalize the quadratic forms by finding an orthogonal matrix \(Q\) such that the change of variable \(\mathbf{x}=\) Qy transforms the given form into one with no cross-product terms. Give Q and the new quadratic form. $$x_{1}^{2}+x_{2}^{2}+3 x_{3}^{2}-4 x_{1} x_{2}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.