Week 8 - Orthogonal diagonalisation, Quadratic forms Flashcards

1
Q

An orthogonal set of non-zero vectors is…

A

linearly independent!

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2
Q

An nxn matrix P is orthogonal if…

A

P^T P = In (identity matrix)

The columns of P form an orthogonal set of UNIT VECTORS

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3
Q

Theorem / 3 checks for orthogonal diagonalisation

A
  1. P^T P = In
  2. P^-1 = P^T (invertible and = identity matrix)
  3. P^T AP = D (diagonal matrix)
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4
Q

Condition for a square matrix to be orthogonally diagonalisable

A

If and only if it is SYMMETRIC, ie. A^T = A

Need an orthogonal matrix P such that P^T AP = D where D is a diagonal matrix

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5
Q

How to solve/sketch the contour for a quadratic form x^T Ax =2?

A
  1. Let x = Pz where z = (X Y)
  2. z^T Dz = {expand & write out full quadratic form}
  3. Determine if ELLIPSE or HYPERBOLA
  4. Find the unit vector in the big X and big Y directions & sketch the X and Y axes (on top of the x and y axes)
  5. Find the X and Y intercepts
  6. Connect the points
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6
Q

How to test for semidefiniteness?

A
  • If A has an eigenvalue = 0, |A|=0 so principal minors test fails.
  • Must find the eigenvalues of A and get >=1 zero evalue
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7
Q

How to show that B^T B is symmetric?

[2022 IRDAP paper]

A

B^T B is symmetric as (B^T B)^T = B^T B

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