Square Roots of Positive Operators
In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) T on a complex Hilbert space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B. According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T½ such that T½ is itself positive and (T½)2 = T. The operator T½ is the unique non-negative square root of T.
A bounded non-negative operator on a complex Hilbert space is self adjoint by definition. So T = (T½)* T½. Conversely, it is trivially true that every operator of the form B* B is non-negative. Therefore, an operator T is non-negative if and only if T = B* B for some B (equivalently, T = CC* for some C).
The Cholesky factorization provides another particular example of square root, which should not be confused with the unique non-negative square root.
Read more about this topic: Square Root Of A Matrix
Famous quotes containing the words square, roots and/or positive:
“I would say it was the coffin of a midget
Or a square baby
Were there not such a din in it.”
—Sylvia Plath (1932–1963)
“Our roots are in the dark; the earth is our country. Why did we look up for blessing—instead of around, and down? What hope we have lies there. Not in the sky full of orbiting spy-eyes and weaponry, but in the earth we have looked down upon. Not from above, but from below. Not in the light that blinds, but in the dark that nourishes, where human beings grow human souls.”
—Ursula K. Le Guin (b. 1929)
“The learner always begins by finding fault, but the scholar sees the positive merit in everything.”
—Georg Wilhelm Friedrich Hegel (1770–1831)