Question

show that the matrix $A$ is positive definite first by using Theorem 7.3 .2 and then by using Theorem 7.3 .4. (a) $A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$ (b) $A=\left[\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right]$

Name: Problem 26, Chapter 7: Elementary Linear Algebra: Applications Version
Uploaded: 2024-02-10T19:31:42-08:00
Channel: Victor Salazar
Description: show that the matrix A is positive definite first by using Theorem 7.3 .2 and then by using Theorem 7.3 .4. (a) A=[ 2 1 1 2 ] (b) A=[ 3 -1 0 -1 2 -1 0 -1 3 ]

   show that the matrix $A$ is positive definite first by using Theorem 7.3 .2 and then by using Theorem 7.3 .4.
(a) $A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
(b) $A=\left[\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right]$

Elementary Linear Algebra: Applications Version

Howard Anton, Chris… 11th Edition

Chapter 7, Problem 26

Instant Answer

Solved by Expert Victor Salazar

02/10/2024

Step 1

3.2, we need to show that all the leading principal minors of $A$ are positive. The leading principal minors of $A$ are: $M_1 = 2$ $M_2 = \left|\begin{array}{cc}2 & 1 \\ 1 & 2\end{array}\right| = 3$ Since both $M_1$ and $M_2$ are positive, we can conclude that Show more…

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show that the matrix $A$ is positive definite first by using Theorem 7.3 .2 and then by using Theorem 7.3 .4. (a) $A=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$ (b) $A=\left[\begin{array}{rrr}3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3\end{array}\right]$

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Key Concepts

Positive Definiteness

This is a property of symmetric matrices where, for every nonzero vector x, the quadratic form x?Ax is positive. It is essential in many areas of mathematics including optimization and statistics, as it guarantees that the matrix induces a strictly convex quadratic form.

Sylvester's Criterion

Sylvester's Criterion is a method for testing the positive definiteness of a symmetric matrix by checking that all of its leading principal minors are positive. This approach simplifies the verification process by breaking the matrix into smaller submatrices.

Eigenvalue Characterization

The eigenvalue characterization of positive definiteness states that a symmetric matrix is positive definite if all its eigenvalues are positive. This connection to the matrix's spectrum is fundamental in linear algebra and aids in understanding the behavior of quadratic forms.

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