Linear Algebra: Symmetric Matrices Quiz

Questions and Answers

What characterizes the set Sn+ of matrices?

All matrices in Sn+ are symmetric and invertible.

All matrices in Sn+ satisfy the condition zTXz ≥ 0 for all vectors z. (correct)

All matrices in Sn+ are singular.

All matrices in Sn+ have at least one negative eigenvalue.

Which of the following statements is true regarding Sn++?

Matrices in Sn++ are always symmetric but not necessarily positive semidefinite.

All matrices in Sn++ can have zero as an eigenvalue.

Sn++ is a subset of Sn+. (correct)

Matrices in Sn++ are characterized by having positive eigenvalues. (correct)

Why is Sn+ considered a convex cone?

It has matrices that do not change under transposition.

It contains only positive definite matrices.

It consists solely of symmetric matrices.

It is closed under scalar multiplication and addition of its matrices. (correct)

Which of these matrices example does belong to S2+?

The matrix with components (2, 1; 1, 2). Signup and view all the answers

What is the condition for a matrix to be in Sn+?

It must be symmetric and satisfy zTXz ≥ 0 for all vectors z. Signup and view all the answers

What is the significance of the parameter θ in the expression $x = θx1 + (1 − θ)x2$?

It defines the specific location of point x on the line. Signup and view all the answers

Under what condition is a set considered an affine set?

It includes all points on the line between two distinct points. Signup and view all the answers

Which value of θ would place the point x exactly at x1?

0 Signup and view all the answers

What relationship exists between an affine set and the solution set of linear equations?

Any affine set corresponds to a solution set of a linear equation system. Signup and view all the answers

Which of the following values for θ represents a point outside the segment between x1 and x2?

−0.2 Signup and view all the answers

Study Notes

Symmetric Matrices

Sn represents the set of symmetric n × n matrices.
Symmetric matrices are those where the transpose equals the original matrix (X = X^T).

Positive Semidefinite Matrices

Sn+ comprises positive semidefinite n × n matrices (X ∈ Sn | X ⪰ 0).
A matrix X is considered positive semidefinite if z^T X z ≥ 0 for all vectors z.
This condition implies that the quadratic form defined by the matrix X is non-negative.

Convex Cone

Sn+ forms a convex cone, meaning that if two positive semidefinite matrices are combined through linear combinations, the result will also be a positive semidefinite matrix.

Positive Definite Matrices

Sn++ denotes the set of positive definite n × n matrices (X ∈ Sn | X ≻ 0).
A positive definite matrix is more restrictive than a semidefinite one and satisfies z^T X z > 0 for all non-zero vectors z.

Example

An example of a positive semidefinite matrix in S2+ is {{x, y}, {y, z}}, where elements must satisfy the positive semidefinite condition.

Line through Points x1 and x2

The equation representing points along the line between two points (x_1) and (x_2) is given by (x = \theta x_1 + (1 - \theta)x_2), where (\theta) is a real number.
The parameter (\theta) affects the position of point (x) on the line:
- When (\theta = 1.2), point (x) is beyond (x_2).
- When (\theta = 1), point (x) coincides with (x_2).
- When (\theta = 0.6), point (x) is positioned near both (x_1) and (x_2) but closer to (x_2).
- When (\theta = 0), point (x) coincides with (x_1).
- When (\theta = -0.2), point (x) is extended beyond (x_1).

Affine Sets

An affine set is defined as containing the line through any two distinct points within the set.
Example of an affine set includes solution sets to linear equations represented as: ({x | Ax = b}).
Each affine set corresponds to a solution set of a system of linear equations, showcasing their dual relationship.

Description

Test your knowledge on symmetric matrices and their properties, focusing on positive semidefinite and positive definite matrices. This quiz covers key concepts such as convex cones and criteria for matrix positivity with examples. Perfect for students studying linear algebra and matrix theory.