Unformatted text preview: Symmetric matrices and diagonalization
Eigenvalues are always real Eigenvalues n LI eigenvectors can be orthogonalized LI A = PDP1 becomes A = QDQT (no need PDP becomes for inverse any more!) Spectral decomposition (spectrum and Spectral rank one matrices) Positive definite matrix and quadratic Positive form xTAx
55 55 Examples (HKAL Pure Math papers?)
What is the relevance? (Background, skill – What good revision is helpful. Eigenvector interpretation for alternatives ) 2007 Paper 1 Q5 2007  Eigenvalues are α and β (why? Similarity and (why? matrices)  Cayley Hamilton (tied up CE with matrix M)  Two determinant properties (what are those?)
56 HKAL 2005 Q8
CE of a 2x2 matrix (trace and det) CE CayleyHamilton to relate CE to matrix Cayley How to get Mn from M by iteratively How multiplying M (observe and practise the skill) α and β are the distinct eigenvalues and (why?) Finding the n power of a 2x2 matrix Finding (worked out example for a more general 57 57 alternative) Matrix power (or power matrix)
How to find An using diagonalization How using A = PDP1 Observe what happen to A2=AA and Observe deduce An. Why finding Dn is easy? Compare with HKAL solution Compare (without using eigenvector interpretation)
58 58 Example on Markov
The important role of the eigenvalue 1 The in the steadystate solution How to find x(t+n) = An x(t) and what will How x(t happen in the steady state when n becomes very large Two ways: using power matrix or using Two Ax=λx (How? x(t) can be expressed as Ax= (How? a linear combination of the eigenvectors) 59 59 More examples
Handling a series or sequence: Handling
• Fibonacci numbers 1,1,2,3,5,8,13,21,… • Time series Difference equation in matrix form Difference
• Equation order and matrix order • Characteristic equation Differential equation in matrix form Differential
• From discretetime to continuoustime • Characteristic equation
60 Linear transformation
Definition of T: T(au+bv) = aT(u)+bT(v) Definition Mapping without matrices Mapping
• Domain and image, range and kernel, 11 and 1onto • Is translation a linear transformation? Transformation matrix A Transformation
• T(x) as matrix multiplication Ax T(x) • Attaching coordinates to basis vectors • A is well defined when the images of all the basis vectors are known
61 61 Change of basis
Similarity transformation Similarity Ax=y, and A = PDP1 Ax=y, Ax=y, and A = QDQT Ax=y, Quadratic form xTAx can become vTDv Quadratic where v= QTx (or Qv=x) How should we interpret change of How basis? Pv=x and Qv=x
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 Spring '10
 Chan
 Linear Algebra, Cayley Hamilton, power matrix, matrix form Difference, matrix multiplication Ax

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