Lecture 15 Symmetric matrices, quadratic forms, matrix ...

[Pages:32]EE263 Autumn 2007-08

Stephen Boyd

Lecture 15

Symmetric matrices, quadratic forms, matrix norm, and SVD

? eigenvectors of symmetric matrices ? quadratic forms ? inequalities for quadratic forms ? positive semidefinite matrices ? norm of a matrix ? singular value decomposition

15?1

Eigenvalues of symmetric matrices

suppose A Rn?n is symmetric, i.e., A = AT

fact: the eigenvalues of A are real to see this, suppose Av = v, v = 0, v Cn

then

n

vT Av = vT (Av) = vT v = |vi|2

i=1

but also

n

vT Av = (Av)T v = (v)T v = |vi|2

i=1

so we have = , i.e., R (hence, can assume v Rn)

Symmetric matrices, quadratic forms, matrix norm, and SVD

15?2

Eigenvectors of symmetric matrices

fact: there is a set of orthonormal eigenvectors of A, i.e., q1, . . . , qn s.t. Aqi = iqi, qiT qj = ij in matrix form: there is an orthogonal Q s.t.

Q-1AQ = QT AQ =

hence we can express A as

n

A = QQT = iqiqiT

i=1

in particular, qi are both left and right eigenvectors

Symmetric matrices, quadratic forms, matrix norm, and SVD

15?3

replacements

Interpretations

A = QQT

x

QT x QT

QT x

Q

Ax

linear mapping y = Ax can be decomposed as ? resolve into qi coordinates ? scale coordinates by i ? reconstitute with basis qi

Symmetric matrices, quadratic forms, matrix norm, and SVD

15?4

or, geometrically, ? rotate by QT ? diagonal real scale (`dilation') by ? rotate back by Q

decomposition

n

A = iqiqiT

i=1

expresses A as linear combination of 1-dimensional projections

Symmetric matrices, quadratic forms, matrix norm, and SVD

15?5

example:

A= =

-1/2 3/2 3/2 -1/2

1 2

11 1 -1

10 0 -2

1 2

11 1 -1

T

q1 q2qx2T x

q1q1T x q2

1q1q1T x

Symmetric matrices, quadratic forms, matrix norm, and SVD

Ax2q2q2T x

15?6

proof (case of i distinct) suppose v1, . . . , vn is a set of linearly independent eigenvectors of A:

Avi = ivi,

vi = 1

then we have

viT (Avj) = jviT vj = (Avi)T vj = iviT vj

so (i - j)viT vj = 0 for i = j, i = j, hence viT vj = 0

? in this case we can say: eigenvectors are orthogonal ? in general case (i not distinct) we must say: eigenvectors can be

chosen to be orthogonal

Symmetric matrices, quadratic forms, matrix norm, and SVD

15?7

Example: RC circuit

i1 v1 c1

resistive circuit in vn cn

ckvk = -ik, i = Gv

G = GT Rn?n is conductance matrix of resistive circuit

thus v = -C-1Gv where C = diag(c1, . . . , cn) note -C-1G is not symmetric

Symmetric matrices, quadratic forms, matrix norm, and SVD

15?8

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