Positive Definite Matrices

Chapter 8 • Section 8-3

"The 'positive numbers' of the matrix world. ➕"

✅ What is Positive Definite?

A symmetric matrix $A$ is positive definite if:

$$ \vec{x}^T A \vec{x} > 0 \quad \text{for all } \vec{x} \neq \vec{0} $$

This means the quadratic form $q(\vec{x}) = \vec{x}^T A \vec{x}$ is always positive, like a bowl shape opening upwards.

Eigenvalue Test

All eigenvalues of $A$ are strictly positive ($\lambda_i > 0$).

Determinant Test

All leading principal submatrices have positive determinants.

Principal Submatrices Lemma

If $A$ is positive definite, then every principal submatrix of $A$ is also positive definite.

This means any square block centered on the diagonal must be PD.

Definiteness Checker

Enter a $2 \times 2$ symmetric matrix to check its definiteness.
$A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$

=

(Symmetric: $a_{12} = a_{21} = b$)

Positive Definite

Eigenvalues: 3.00, 1.00 (Both > 0)
Shape: Bowl (Minimum at 0)

🥣

Cholesky Factorization

Every positive definite matrix $A$ can be factored uniquely as:

$$ A = U^T U $$

Where $U$ is an upper triangular matrix with positive diagonal entries. This is like the "square root" of a matrix!

Algorithm (Row Operations)

We can find $U$ by reducing $A$ to upper triangular form using only row operations of type 3 (adding a multiple of one row to another), but with a twist: we do the same operation to columns!

  1. Perform row operations to zero out entries below the pivot.
  2. Simultaneously perform the same column operations to zero out entries to the right of the pivot.
  3. Since $A$ is symmetric, this preserves symmetry.
  4. The resulting diagonal matrix $D$ will have positive entries.
  5. $U = \sqrt{D} L^{-1}$, where $L$ tracks the operations. (Or simply use the formula $u_{ii} = \sqrt{a_{ii} - \sum u_{ki}^2}$).

🧠 Knowledge Check Win $15

Is the identity matrix $I$ positive definite?