seg2018-6 - Symmetric matrices and diagonalization...

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Unformatted text preview: Symmetric matrices and diagonalization Eigenvalues are always real Eigenvalues n LI eigenvectors can be orthogonalized LI A = PDP-1 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 Cayley-Hamilton 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 = PDP-1 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 steady-state 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 discrete-time to continuous-time • 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, 1-1 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 = PDP-1 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 62 ...
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