Positive Definite Matrices Calculator

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Example

Created on 2024-06-20Asked by Olivia Lopez (Solvelet student)

Determine whether the matrix

A = \begin{bmatrix} 4 & -1 \\ -1 & 3 \end{bmatrix}

is positive definite.

Solution

To determine whether the matrix

A = \begin{bmatrix} 4 & -1 \\ -1 & 3 \end{bmatrix}

is positive definite: 1. Calculate the determinant of

A

\det(A) = \left| \begin{matrix} 4 & -1 \\ -1 & 3 \end{matrix} \right| = (4)(3) - (-1)(-1) = 13.

2. Check if all leading principal minors of

A

are positive. For a

2 \times 2

matrix, this means

\det(A) > 0

\det(A) = 13 > 0.

Since the determinant of

A

is positive, the matrix

A

is positive definite. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Alexander Lopez on Solvelet

1. Determine whether the matrix

A = \begin{bmatrix} 4 & 1 \\ 1 & 3 \end{bmatrix}

is positive definite.2. Find the eigenvalues of the matrix

B = \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}

and determine its positive definiteness.

DefinitionSo we have positive definite matrix A if it is symmetric, i.e. if (A)^T = A for all whose eigenvalues are greater than zero. That is, the quadratic form xTAx>0 for all non-zero x. Optimization and numerical analysis Positive definite StringIO matrices are a key ingredient in optimization and numerical analysis. For example, well-known ar positive definite where B=⟨2112⟩.

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