Which of the following 2 ×2 matrices are positive definite?...

Which of the following $2 \times 2$ matrices are positive definite? (a) $\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right)$, (b) $\left(\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right)$, (c) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$, (d) $\left(\begin{array}{rr}5 & 3 \\ 3 & -2\end{array}\right)$, (e) $\left(\begin{array}{rr}1 & -1 \\ -1 & 3\end{array}\right)$ (f) $\left(\begin{array}{rr}1 & 1 \\ -1 & 2\end{array}\right)$ In the positive definite cases, write down the formula for the associated inner product.

Which of the following $2 \times 2$ matrices are positive definite?
(a) $\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rr}5 & 3 \\ 3 & -2\end{array}\right)$,
(e) $\left(\begin{array}{rr}1 & -1 \\ -1 & 3\end{array}\right)$
(f) $\left(\begin{array}{rr}1 & 1 \\ -1 & 2\end{array}\right)$
In the positive definite cases, write down the formula for the associated inner product.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Matrix-Induced Inner Product

When a symmetric matrix is positive definite, it can be used to define an inner product on a vector space via the formula ?x, y? = x?Ay. This induced inner product retains the properties of symmetry, linearity, and positive definiteness, and provides a useful tool in various applications, such as defining norms and distances in the space.

Positive Definiteness

Positive definiteness is a property of matrices which guarantees that for any non-zero vector x, the quadratic form x?Ax is strictly positive. For real symmetric matrices, this condition is equivalent to expecting all eigenvalues to be positive. This concept is central in many areas, including optimization and statistics, ensuring stability and the uniqueness of solutions in systems involving such matrices.

Sylvester's Criterion

Sylvester’s criterion provides a systematic way to determine the positive definiteness of a symmetric matrix by checking if all the leading principal minors (determinants of the upper-left k×k submatrices) are positive. In a 2×2 case, it requires that the top left element is positive and that the determinant of the matrix is positive.

Transcript

Please give Ace some feedback