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Chapter 4: Problem 86

Let \(A\) be an \(n \times n\) symmetric matrix, such that the quadratic form. \(Q_{A}(\vec{x})=\vec{x} \cdot A \vec{x}\) is positive definite. What is the volume of the region \(Q(\vec{x}) \leq 1 ?\)

Short Answer

Expert verified
\( \frac{\pi^{n/2}}{\Gamma(n/2 + 1) \sqrt{\text{det}(A)}} \)

Step by step solution

01

Understanding the Quadratic Form

The quadratic form is given by \( Q_{A}(oldsymbol{x}) = oldsymbol{x} oldsymbol{ extbf{A}} oldsymbol{x} \), where \( oldsymbol{ extbf{A}} \) is a symmetric \( n \times n \) matrix. We need to find the volume of the region where \( Q(oldsymbol{x}) \leq 1 \), which is a general ellipsoid in \( n \) dimensions.
02

Diagonalization of Matrix A

Since \( oldsymbol{ extbf{A}} \) is symmetric, it can be diagonalized. There exists an orthogonal matrix \( oldsymbol{ extbf{P}} \) such that \( oldsymbol{ extbf{P}}^{T} oldsymbol{ extbf{A}} oldsymbol{ extbf{P}} = oldsymbol{ extbf{D}} \) where \( oldsymbol{ extbf{D}} \) is a diagonal matrix with eigenvalues of \( oldsymbol{ extbf{A}} \).
03

Change of Variables

Let \( oldsymbol{y} = oldsymbol{ extbf{P}}^{T} oldsymbol{x} \). Then \( Q_{A}(oldsymbol{x}) = oldsymbol{x} oldsymbol{ extbf{A}} oldsymbol{x} = oldsymbol{y} oldsymbol{ extbf{D}} oldsymbol{y} \). In terms of \( oldsymbol{y} \), the ellipsoid \( Q_{A}(oldsymbol{x}) \leq 1 \) becomes \( oldsymbol{y} oldsymbol{ extbf{D}} oldsymbol{y} \leq 1 \).
04

Analyzing the Transformed Inequality

Since \( oldsymbol{ extbf{D}} \) is diagonal, \( oldsymbol{y} oldsymbol{ extbf{D}} oldsymbol{y} = rac{y_1^2}{ rac{1}{ ext{eigenvalue } \lambda_1}} + rac{y_2^2}{ rac{1}{ ext{eigenvalue } \lambda_2}} + ext{...}+ \frac{y_n^2}{ rac{1}{ ext{eigenvalue } \lambda_n}} \leq 1 \). This represents an ellipsoid with axes scaled by square roots of eigenvalues.
05

Calculating the Volume of the Ellipsoid

The volume \( V \) of an ellipsoid defined by \( \sum_{i=1}^{n} \frac{y_i^2}{a_i^2} \leq 1 \) where each \( a_i = \sqrt{1/\lambda_i} \) is given by the formula \( \frac{\pi^{n/2}}{\Gamma(n/2 + 1)} \prod_{i=1}^{n} a_i \).
06

Relating Back to Original Variables

Substituting \( a_i = \sqrt{1/\lambda_i} \), the volume \( V = \frac{\pi^{n/2}}{\Gamma(n/2 + 1)} \prod_{i=1}^{n} \sqrt{1/\lambda_i} \). This is equivalent to \( V = \frac{\pi^{n/2}}{\Gamma(n/2 + 1) \sqrt{\text{det}(A)}} \)
07

Conclusion

The volume of the region defined by \( Q_{A}(oldsymbol{x}) \leq 1 \) is \( \frac{\pi^{n/2}}{\Gamma(n/2 + 1) \sqrt{\text{det}(A)}} \).

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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 equal to its transpose. This means if you flip the matrix over its diagonal, it looks the same. Symmetric matrices play a significant role in various mathematical areas, particularly linear algebra and geometry.
They are important because:
  • They simplify many calculations, particularly with eigenvalues and eigenvectors.
  • The quadratic forms associated with symmetric matrices can be easily analyzed.
  • Spectral theorem: every symmetric matrix can be diagonalized using an orthogonal matrix, which simplifies complexity in matrix equations.
Symmetric matrices are often related to real-world symmetries, making them widely applicable in physics, engineering, and computer science.
Quadratic Form
A quadratic form is a polynomial with terms that are all of degree two. In the context of matrices, a quadratic form in variables is expressed as \(Q(\vec{x}) = \vec{x}^T A \vec{x}\), where \(A\) is a symmetric matrix.
This form is essential in defining geometric figures, particularly ellipsoids and other multidimensional shapes. Quadratic forms are crucial for:
  • Understanding properties of graphs related to curvature and surface analysis.
  • Identifying the nature of conic sections and their high-dimensional analogs.
  • Providing insights into stability within physics and optimizing multivariable systems.
They are initially abstract but unravel numerous practical applications when analyzed with matrices.
Ellipsoid
An ellipsoid is the set of all points in a space where the sum of squared distances from each point to the center is constant. It generalizes the notion of an ellipse to higher dimensions. Mathematically, in a space of dimension \(n\), an ellipsoid is represented by \(\sum_{i=1}^{n} \frac{x_i^2}{a_i^2} = 1\), where all \(a_i\) are positive.
When talking about matrices, specifically symmetric ones, ellipsoids are represented by quadratic forms such as \(Q_A(\vec{x}) \leq 1\).
Ellipsoids are significant because they have:
  • Applications in optimization, as they can represent constraints or objective regions.
  • Importance in statistics as multi-dimensional normal distribution can have ellipsoidal contours.
  • Relevance in mechanics where they represent cross-sections of stress within materials.
Understanding ellipsoids helps visualize data trends and physical phenomena in multi-dimensional spaces.
Eigenvalues
Eigenvalues are scalars associated with a square matrix, indicating how a transformation represented by that matrix stretches or shrinks vectors. For a given square matrix \(A\), if \(\vec{v}\) is a non-zero vector, an eigenvalue \(\lambda\) satisfies the equation \(A\vec{v} = \lambda \vec{v}\).
In symmetric matrices, eigenvalues have special properties:
  • They are always real numbers, which simplifies analysis in real vector spaces.
  • They can determine definiteness of a matrix, indicating if a quadratic form is positive definite.
  • Associated eigenvectors of distinct eigenvalues are orthogonal, aiding in matrix diagonalization.
The role of eigenvalues is pivotal in stability analysis in engineering, vibration modes in mechanical systems, and dimensionality reduction techniques like Principal Component Analysis in statistics.
Orthogonal Matrix
An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning every vector is perpendicular to each other and has a unit length. A matrix \(P\) is orthogonal if its transpose equals its inverse, or \(P^T P = I\), where \(I\) is the identity matrix.
Orthogonal matrices have valuable properties:
  • Preserving lengths and angles, meaning they represent rigid transformations like rotations and reflections.
  • Aiding matrix diagonalization, crucial for simplifying complex linear transformations.
  • Possessing det(P) = \(\pm 1\), emphasizing non-distorting transformations in geometric interpretations.
Orthogonal matrices appear in numerous applications, from simplifying matrix computations in numerical analysis to formulating fundamental mechanics and algorithms in computer graphics.

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Most popular questions from this chapter

What is the volume of each of the following dyadic cubes? What dimension is the volume (i.e., are the cubes two-dimensional, three-dimensional or what)? What information is given below that you don't need to answer those two questions? (a) \(C_{\left[\begin{array}{l}1 \\ 2\end{array}\right], 33\) (b) \(C_{\left[\begin{array}{l}0 \\ 1 \\ 3\end{array}\right], 22\) (c) \(C_{\left[\begin{array}{l}0 \\ 1 \\ 1 \\ 1\end{array}\right] \cdot 3} \end{array}\) (d) \(C\left[\begin{array}{l}0 \\ 1 \\ 4\end{array}\right] .3\)

What elementary matrices are permutation matrices? Describe the corresponding permutation.

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