MATLAB: How to generate B-orthonormal matrix X, which means X’BX=I

b-orthonormal

Now I need to generate the initial value for random matrix X, which satisfies X'*B*X=I (not orthogonal X'*X=I). B is Hermitian positive. It is called B-orthonormal. But I don't know how to do it, I only know "orth" can find X'*X=I. But what I need is B-orthonormal, X'*B*X=I. Anyone knows can to do it in Matlab? Thank you in advance!

Best Answer

% Assuming B is Hermitian sdp
n = length(B);
[Q,~] = qr(randn(n));
C = chol(B)
X = C\Q
% Check
X'*B*X

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function plot doesn't work with input type syms

Instead try the following

x=[-10:0.0001:10];
y=exp(x)./x;
plot(x,y);
axis([-10 10 -1e3 1e3])
grid on
title('e^x/x')

and the following doesn't crash

syms x x0;
  y= exp(x-x0)/(x-x0)
  x_tay4=taylor(y, 'order',4)
  y =
  exp(x - x0)/(x - x0)
  x_tay4 =
  (- exp(-x0)/(6*x0) - exp(-x0)/(2*x0^2) - exp(-x0)/x0^3 - exp(-x0)/x0^4)*x^3 + (- exp(-x0)/(2*x0) - exp(-x0)/x0^2 - exp(-x0)/x0^3)*x^2 + (- exp(-x0)/x0 - exp(-x0)/x0^2)*x - exp(-x0)/x0

if you find these lines useful would you please mark my answer as Accepted Answer?

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thanks in advance for time and attention

John BG

<mailto:jgb2012@sky.com jgb2012@sky.com>

Best Answer

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