Notes - Continuous Mathematics HT23, Definiteness

Created: January 24, 2023

Flashcards

What does it mean for a symmetric matrix $\mathbf{A}$ to be positive definite?
\[\pmb x^\intercal \mathbf A \pmb x > 0\]

for all nonzero $x$.

What does it mean for a symmetric matrix $\mathbf{A}$ to be positive semidefinite?
\[\pmb x^\intercal \mathbf A \pmb x \ge 0\]

for all nonzero $x$.

What does it mean for a symmetric matrix $\mathbf{A}$ to be negative definite?
\[\pmb x^\intercal \mathbf A \pmb x < 0\]

for all nonzero $x$.

What does it mean for a symmetric matrix $\mathbf{A}$ to be negative semidefinite?
\[\pmb x^\intercal \mathbf A \pmb x \le 0\]

for all nonzero $x$.

What does it mean for a symmetric matrix $\mathbf{A}$ to be indefinite?

$\mathbf{A}$ is not positive semidefinite or negative semidefinite.

\[\mathbf A = \begin{pmatrix} a \& b \\\\b\&c \end{pmatrix}\]

What conditions are there for $\mathbf A$ being positive definite, postivie semidefinite, negative definite, negative semidefinite and indefinite in terms of the entries $a, b, c$?
  • Positive definite iff $ \vert \mathbf A \vert > 0$ and $a > 0$.
  • Positive semidefinite (and not definite) iff $ \vert \mathbf A \vert = 0$ and $a + c \ge 0$.
  • Negative definite iff $ \vert \mathbf A \vert > 0$ and $a < 0$.
  • Negative semidefinite (and not definite) iff $ \vert \mathbf A \vert = 0$ and $a + c \le 0$.
  • Indefinite iff $ \vert \mathbf{A} \vert < 0$.

If the eigenvalues of $A$ are $\lambda _ 1, \ldots, \lambda _ n$, what is $\det A$?
\[\prod_{i=1}^n \lambda_i\]

If the eigenvalues of $A$ are $\lambda _ 1, \ldots, \lambda _ n$, what is $\text{tr} (A)$?
\[\sum _ {i=1}^n \lambda _ i\]

If

\[\mathbf A = \begin{pmatrix} a \& b \\\\b\&c \end{pmatrix}\]

then where do the rules for $\mathbf A$ being definite

  • Positive definite iff $ \vert \mathbf A \vert > 0$ and $a > 0$.
  • Positive semidefinite (and not definite) iff $ \vert \mathbf A \vert = 0$ and $a + c \ge 0$.
  • Negative definite iff $ \vert \mathbf A \vert > 0$ and $a < 0$.
  • Negative semidefinite (and not definite) iff $ \vert \mathbf A \vert = 0$ and $a + c \le 0$.
  • Indefinite iff $ \vert \mathbf{A} \vert < 0$.

come from?

The rules about the definiteness of $\mathbf A$ being determined by the sign of all its eigenvalues, and $\det \mathbf{A} = \prod _ {i=1}^n \lambda _ i$ and $\text{tr}(\mathbf A) = \sum _ {i=1}^n \lambda _ i$.

\[\mathbf A = \mathbf A^\intercal\]

What conditions are there for $\mathbf A$ being positive definite, postivie semidefinite, negative definite, negative semidefinite and indefinite in terms of the eigenvalues?
  • Positive definite if all eigenvalues $> 0$.
  • Positive semidefinite if all eigenvalues $\ge 0$.
  • Negative definite if all eigenvalues $< 0$.
  • Negative semidefinite if all eigenvalues $\le 0$.
\[\mathbf A = \mathbf A^\intercal\]

What conditions are there for $\mathbf A$ being positive definite, postivie semidefinite, negative definite, negative semidefinite and indefinite in terms of the pivots?
  • Positive definite if all pivots $> 0$.
  • Positive semidefinite if all pivots $\ge 0$.
  • Negative definite if all pivots $< 0$.
  • Negative semidefinite if all pivots $\le 0$.

What are the pivots of a matrix $\mathbf A$?

The entries on the diagonal in the row-reduced form of $\mathbf A$.

$-\mathbf A$ is negative (semi)definite iff…

$\mathbf A$ is positive semidefinite.

The sum of positive semidefinite matrices is…

Positive semidefinite.

The sum of a positive definite and a positive (semi)definite is…

Positive definite.

If $\mathbf C$ is any symmetric matrix, what is true about the definiteness $\mathbf C - \lambda \mathbf I$?

It is positive semidefinite as long as $\lambda \le$ the smallest eigenvalue of $\mathbf{C}$.

If $\mathbf A$ is a (semi)definite matrix, when is $c\mathbf A$ positive (semi)definite?

$c > 0$.

If $\mathbf{A}$ is a positive (semi)definite matrix, is $\mathbf A^{-1}$ positive semidefinite?

Yes.

If $\mathbf A$ is a positive (semi)definite matrix, what is true about any upper-left submatrix of $\mathbf A$?

It is also positive (semi)definite.

If $\mathbf A, \mathbf B$ are positive (semi)definite matrices, then is $\mathbf A \mathbf B$ positive (semi)definite?

No.

If $\mathbf A, \mathbf B$ are positive (semi)definite matrices, then is $\mathbf{ABA}$ positive (semi)definite?

Yes.

If $A$ is a positive (semi)definite matrix, and $\mathbf C$ is any matrix, what is true about $\mathbf{C}^\intercal\mathbf{AC}$?

It is positive semidefinite.

If $A$ is a positive (semi)definite matrix, and $\mathbf C$ is any matrix, then $\mathbf{C}^\intercal\mathbf{AC}$ is positive semidefinite. What condition allows you to guarantee it will in fact be positive definite?

$\mathbf C$ has full column rank.

In the real numbers, if $A$ is positive, then $A - x$ will still be positive as long as $x < A$. What is a similar property when $\mathbf A$ is a positive definite matrix?
\[\mathbf A - \pmb v \pmb v^\intercal\]

is positive definite if $\pmb v^\intercal \mathbf A^{-1} \pmb v < 1$.

What’s the notation for $\mathbf A$ being positive semidefinite?
\[A \succeq 0\]


Related posts