Linear Algebra: Symmetric Matrices Quiz

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.

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.