Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

• A non-zero vector v in vector space V is an eigenvector of linear transformation T if T(v) = λv, where λ is a scalar.
• λ is the eigenvalue of T corresponding to the eigenvector v.
• If v is an eigenvector, any non-zero scalar multiple of v is also an eigenvector with the same eigenvalue.
• If v₁, v₂,...vₖ are eigenvectors of T corresponding to distinct eigenvalues λ₁, λ₂,...λₖ, then v₁, v₂,...vₖ are linearly independent.
• The set of eigenvectors of T for an eigenvalue λ, along with the zero vector, forms a subspace of V called the eigenspace of T corresponding to λ.

Finding Eigenvalues and Eigenvectors

• Given A as the matrix representation of T, v is an eigenvector of T for eigenvalue λ if and only if (A - λI)v = 0, where I is the identity matrix.

Characteristic Polynomial

• The characteristic polynomial of A is defined as p(λ) = det(A - λI).
• Eigenvalues of A are roots of the characteristic polynomial.

Finding Eigenvectors

• For each eigenvalue λ, eigenvectors are the non-zero solutions to the homogeneous system (A - λI)v = 0.

Clinical Guidelines: Pre-assessment

• Involves gathering patient information through history-taking, observation, palpation, movement assessments, neurological exams, and special tests.

History

• Presenting Complaint: Details about the patient's current issue, including character, onset, etc., using VAS/numerical pain scores.
• Past Medical History: Includes significant medical and surgical information, medications (dose, route, frequency), and allergies.
• Social History: Covers smoking, alcohol, living situation, occupation, and hobbies.
• Functional History: Focuses on ADLs (Activities of Daily Living) and how the complaint affects the patient's life.

Observation

• Gait, posture, scars, swelling, muscle wasting, and skin changes are noted.

Palpation

• Involves assessing skin temperature, tenderness, swelling, muscle spasm, and anatomical landmarks.

Active Movements

• Checks range, pain, symptom reproduction, quality, and willingness.

Passive Movements

• Evaluates range, pain, symptom reproduction, end feel, and capsular pattern.

Resisted Isometric Movements

• Assesses strength and pain during isometric contractions to identify muscle or tendon issues.
• Possible outcomes: Strong and Painless, Strong and Painful, Weak and Painless, Weak and Painful

Neurological Examination

• Sensation: Light touch and sharp/dull discrimination tests.
• Myotomes: Evaluates muscle groups innervated by specific nerve roots.
• Reflexes: Tests include Biceps (C5/6), Brachioradialis (C6), Triceps (C7), Patella (L3/4), and Achilles (S1/2).
• Neural Tension Tests: ULNTT (Upper Limb Neural Tension Tests) and slump test are performed.

Special Tests

• Specific tests are used based on the joint being examined and clinical reasoning.

Additional Information

• Includes X-rays, scans, blood tests, referral letters, and other pertinent information.

Comparison of Complex Numbers

• Given two complex numbers z = a + bi and z' = a' + b'i, where a, a', b, b' are real numbers:
• z = z' if and only if a = a' and b = b'.
• Example: If z = (x + y) + i(x - y) and z' = 2 + 3i, then z = z' if and only if x = 5/2 and y = -1/2.

Modulus of a Complex Number

• Given a complex number z = a + bi where a, b are real numbers:
• The modulus of z, denoted as |z|, is a non-negative real number defined as |z| = √(a² + b²).

Properties of Modulus

• |z| = |-z| = |z̄| (where z̄ is the complex conjugate of z)
• |zz'| = |z||z'|
• |zⁿ| = |z|ⁿ for all n ∈ ℕ
• |z/z'| = |z|/|z'| if z' ≠ 0
• |z + z'| ≤ |z| + |z'| (triangle inequality)
• ||z| - |z'|| ≤ |z - z'|

Examples

• If z = 3 + 4i, then |z| = √(3² + 4²) = 5.
• If z = 5 - 12i, then |z| = √(5² + (-12)²) = 13.