Home Features Data Analytics Data Analytics: Summaries on Strang's Linear Algebra - Positive Definite Matrices
Data Analytics: Summaries on Strang's Linear Algebra - Positive Definite Matrices PDF Print E-mail
Written by FemiByte   
Sunday, 10 March 2013 03:24

Positive Definite Matrices

Positive definite matrices are symmetric matrices that have positive eigenvalues.

Tests for positive-definiteness

If a symmetric matrix has 1 of these 5 properties, it has them all:

  1. All $n$ pivots are positive
  2. All $n$ upper left determinants are positive.
  3. All $n$ eigenvalues are positive.
  4. $A$ is positive definite if $x^TAx > 0$ for every nonzero vector $x$. This is the energy-based definition
  5. If the columns of $R$ are independent, then $A=R^TR$ is positive definite. $R$ is a rectangular matrix.

Calculus: $min\text{ }f(x)$ occurs when $df/dx=0$ and $d^2f/dx^2=0$

Linalg: $min\text{ }f(x,y)=x^TAx$ occurs when $\left[ \begin{array}{cc}\delta^2f/\delta x^2&&\delta^2f/\delta x \delta y\\\delta^2f/\delta y\delta x&&\delta^2f/\delta y^2 \end{array}\right]$ is +ve definite.

Suppose $A=Q\Lambda Q^T$ is +ve definite, so $\lambda_i > 0$. The graph of $x^TAx=1$ is ellipse: \[ \left[x\text{ }y \right]Q\Lambda Q^T \left[\begin{array}{c}x\\y\end{array} \right] = \left[X\text{ }Y \right]\Lambda \left[\begin{array}{c}X\\Y\end{array} \right] = \lambda_1 X^2 + \lambda_2 Y^2 =1 \] The axes point along eigenvectors. The half-lengths are $1/\sqrt{\lambda_1}$ and $1/\sqrt{\lambda_2}$.

Key Ideas

  1. Positive definite matrices have positive eigenvalues and positive pivots.
  2. A quick test is given by the upper left determinants: $a>0$ and $ac-b^2 > 0$.
  3. The graph of $x^TAx$ is then a "bowl" going up from $x=0$:
    $x^TAx=ax^2+2bxy+cy^2$is positive except at $(x,y)=(0,0).$
  4. $A=R^TR$ is automatically positive definite $R$ has independent columns.
  5. The ellipse $x^TAx=1$ has its axes along the eigenvectors of $A$. Lengths $1/\sqrt{\lambda}$.
Last Updated on Tuesday, 19 March 2013 21:54
 

joomla 1.5 stats