Chapter 8: Problem 8
Let \(A_{0}\) be formed from \(A\) by deleting rows 2 and 4 and deleting columns 2 and \(4 .\) If \(A\) is positive definite, show that \(A_{0}\) is positive definite.
Short Answer
Expert verified
\(A_0\) is positive definite because it is a submatrix of the positive definite matrix \(A\).
Step by step solution
01
Understanding Positive Definiteness
A matrix is positive definite if it's symmetric and all its leading principal minors are positive. This means that for a matrix to be positive definite, it must be square and symmetric, and each of its upper left submatrices (leading principal submatrices) must have a positive determinant.
02
Structure of Matrix A
Given that matrix \(A\) is positive definite, it satisfies all conditions of positive definiteness. Matrix \(A\) is symmetric and all its leading principal minors are positive.
03
Construction of Submatrix \(A_0\)
Matrix \(A_0\) is created by removing the 2nd and 4th rows and columns from matrix \(A\). Hence, \(A_0\) is a submatrix of \(A\).
04
Applying Positive Definiteness to \(A_0\)
Since \(A\) is positive definite, all minors of \(A\), including those formed by removing any set of rows and corresponding columns, should be positive. This means the minor that corresponds to \(A_0\), formed by deleting rows and columns, remains positive definite.
05
Conclusion on Positive Definiteness of \(A_0\)
Given that the submatrix \(A_0\) is derived from the positive definite matrix \(A\), and the properties of positive definiteness ensure that any such submatrix retains this property, \(A_0\) is also positive definite.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetric Matrices
Symmetric matrices are a special class of square matrices that are equal to their transpose. This means that the entries of the matrix are mirrored along its main diagonal. So, for a symmetric matrix, the element located at the \(i, j\) position is the same as the element at the \(j, i\) position.
For example, if a matrix is symmetric, then \(A_{ij} = A_{ji}\) for every i and j. This property of symmetry is essential because it ensures various mathematical conditions that help in analyzing and solving problems in linear algebra and beyond.
Symmetric matrices arise naturally in many applications such as in structural analysis, data covariance analysis, and optimization problems. They are especially significant in defining positive definite matrices, where the symmetric property is a prerequisite for establishing positive definiteness.
Additionally, due to their symmetry, operations on symmetric matrices can be simplified, often reducing computational complexity. Their eigenvalues are real, and they can be orthogonally diagonalized, which is another valuable property in many calculations.
For example, if a matrix is symmetric, then \(A_{ij} = A_{ji}\) for every i and j. This property of symmetry is essential because it ensures various mathematical conditions that help in analyzing and solving problems in linear algebra and beyond.
Symmetric matrices arise naturally in many applications such as in structural analysis, data covariance analysis, and optimization problems. They are especially significant in defining positive definite matrices, where the symmetric property is a prerequisite for establishing positive definiteness.
Additionally, due to their symmetry, operations on symmetric matrices can be simplified, often reducing computational complexity. Their eigenvalues are real, and they can be orthogonally diagonalized, which is another valuable property in many calculations.
Leading Principal Minors
Leading principal minors are the determinants of the leading principal submatrices of a square matrix. A leading principal submatrix is formed by deleting certain rows and columns consistently from the bottom right side of the matrix.
For instance, consider a matrix of size \( n \times n \). Its leading principal minors are derived from the submatrices of size \(k \times k\) where \( k = 1, 2, \ldots, n \.\) These submatrices are formed by maintaining the top-left \(k \times k\) block of the original matrix.
The significance of leading principal minors comes into play when discussing the positive definiteness of matrices. For a matrix to be positive definite, not only must it be symmetric, but all of its leading principal minors must have positive determinants.
This condition ensures that all potential submatrices have positive determinants, which reflects the absence of zero or negative inner products in applications such as physics or data science, where matrices often represent data or energy states.
For instance, consider a matrix of size \( n \times n \). Its leading principal minors are derived from the submatrices of size \(k \times k\) where \( k = 1, 2, \ldots, n \.\) These submatrices are formed by maintaining the top-left \(k \times k\) block of the original matrix.
The significance of leading principal minors comes into play when discussing the positive definiteness of matrices. For a matrix to be positive definite, not only must it be symmetric, but all of its leading principal minors must have positive determinants.
This condition ensures that all potential submatrices have positive determinants, which reflects the absence of zero or negative inner products in applications such as physics or data science, where matrices often represent data or energy states.
Matrix Submatrices
Matrix submatrices are derived by removing one or more rows and columns from a larger matrix. They can be used to simplify complex matrices into smaller, more manageable pieces.
The procedure of forming submatrices is straightforward: simply eliminate specific rows and columns as desired. This operation is crucial in many areas of linear algebra, particularly when dealing with determinants, eigenvalues, and matrix applications in economics or engineering.
In the context of positive definite matrices, submatrices are of special interest. For instance, if a matrix is positive definite, any submatrix formed by removing the same set of rows and corresponding columns will also inherit the property of positive definiteness, as long as the removal retains the structure required for the definition.
Submatrices can reveal much about the properties of the parent matrix and are fundamental tools when applying results to real-world problems. They help in proving properties such as positive definiteness or exploring specific parts of a matrix, such as those related to block matrices used in partitioned matrix analysis.
The procedure of forming submatrices is straightforward: simply eliminate specific rows and columns as desired. This operation is crucial in many areas of linear algebra, particularly when dealing with determinants, eigenvalues, and matrix applications in economics or engineering.
In the context of positive definite matrices, submatrices are of special interest. For instance, if a matrix is positive definite, any submatrix formed by removing the same set of rows and corresponding columns will also inherit the property of positive definiteness, as long as the removal retains the structure required for the definition.
Submatrices can reveal much about the properties of the parent matrix and are fundamental tools when applying results to real-world problems. They help in proving properties such as positive definiteness or exploring specific parts of a matrix, such as those related to block matrices used in partitioned matrix analysis.
