Eigenvalue Problems. Solving linear systems A x = b is one part of numerical linear algebra, and involves manipulating the rows of a matrix. The second main part of numerical linear algebra is about transforming a matrix to leave its eigenvalues unchanged. A x = x
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Ax = x
where is an eigenvalue of A and non-zero x is the corresponding eigenvector.
a11x1+a12x2+a13x3 = 0
a21x1+a22x2+a23x3 = 0
a31x1+a32x2+a33x3 = 0
(a11a22-a21a12)x2+(a11a23-a21a13)x3 = 0
(a11a32-a31a12)x2+(a11a33-a31a13)x3 = 0
[(a11a22-a21a12)(a11a33-a31a13)- (a11a32-a31a12)(a11a23-a21a13)]x3 = 0
[a11(a22a33 –a23a32) – a12(a21a33-a23a31) + a13(a21a32-a22a31)]x3 = 0
det(A) = a11(a22a33 –a23a32) – a12(a21a33-a23a31) + a13(a21a32-a22a31)
where i can be any value i=1,…n.
det(T) = t11t22…tnn
det(A) = d1d2dn
where the plus or minus sign depends on whether the number of row exchanges is even or odd.
(A-I)x = 0
which holds for x0, so (A- I) is singular and
det(A-I) = 0
xHAx > 0
where xH denotes the conjugate transpose of x (i.e., change the sign of the imaginary part of each component of x and then transpose).
Ax = x xHAx = xHx
A = XX-1
A2 = (XX-1)(XX-1) = X(X-1X)X-1 = X2X-1
Hence we have:
Ap = XpX-1
yk+1 = Ayk
where we start with some initial y0, so that:
yk = Aky0
yk = Aky0 = XkX-1y0
where dot(a,b) is the scalar product of vectors a and b defined by:
dot(a,b) = a1b1+a2b2+…+anbn
yk+1 = A(yk /k+1)
where k+1 is the component of Ayk with largest absolute value.
y = A*y;
[c j] = max(abs(y));
lambda = y(j);
y = y/lambda;
Ax=x TBT-1x=x B(T-1x)=(T-1x)
A = QR
where Q is a matrix whose columns are orthonormal, and R is an upper triangular matrix.
A0 = A
QkRk = Ak
Ak+1 = RkQk
Ak+1 = RkQk = Qk-1AkQk
then Ak and Ak+1 are similar and so have the same eigenvalues.
Ak+1 tends to an upper triangular matrix with the same eigenvalues as A. These eigenvalues lie along the main diagonal of Ak+1.
Ak+1 = RkQk + sI
= Qk-1(Ak – sI)Qk +sI
so once again Ak and Ak+1 are similar and so have the same eigenvalues.
A0 = A
s = Ak(n,n)
QkRk = Ak - sI
Ak+1 = RkQk + sI
The shift operation subtracts s from each eigenvalue of A, and speeds up convergence.
n = size(A,1);
I = eye(n,n);
s = A(n,n); [Q,R] = qr(A-s*I); A = R*Q+s*I