Square root of a matrix | EPFL Graph Search

In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A. Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B). Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in . In general, a matrix can have several square roots. In particular, if then as well. The 2×2 identity matrix has infinitely many square roots. They are given by and where are any numbers (real or complex) such that . In particular if is any Pythagorean triple—that is, any set of positive integers such that , then is a square root matrix of which is symmetric and has rational entries. Thus Minus identity has a square root, for example: which can be used to represent the imaginary unit i and hence all complex numbers using 2×2 real matrices, see Matrix representation of complex numbers. Just as with the real numbers, a real matrix may fail to have a real square root, but have a square root with complex-valued entries. Some matrices have no square root. An example is the matrix While the square root of a nonnegative integer is either again an integer or an irrational number, in contrast an integer matrix can have a square root whose entries are rational, yet non-integral, as in examples above. Positive definite matrix#Decomposition A symmetric real n × n matrix is called positive semidefinite if for all (here denotes the transpose, changing a column vector x into a row vector). A square real matrix is positive semidefinite if and only if for some matrix B. There can be many different such matrices B. A positive semidefinite matrix A can also have many matrices B such that .

Experiments and gyrokinetic simulations of TCV plasmas with negative triangularity in view of DTT operations

Olivier Sauter, Stefano Coda, Justin Richard Ball, Alberto Mariani, Matteo Vallar, Filippo Bagnato

Negative triangularity (NT) scenarios in TCV have been compared to positive triangularity (PT) scenarios using the same plasma shapes foreseen for divertor tokamak test tokamak operations. The experiments provided a NT/PT L-mode pair and a PT H-mode with d ...

Iop Publishing Ltd2024

First-principles thermodynamics of precipitation in aluminum-containing refractory alloys

Anirudh Raju Natarajan

Materials for high -temperature environments are actively being investigated for deployment in aerospace and nuclear applications. This study uses computational approaches to unravel the crystallography and thermodynamics of a promising class of refractory ...

Pergamon-Elsevier Science Ltd2024

SPEEDING UP KRYLOV SUBSPACE METHODS FOR COMPUTING f(A)b VIA RANDOMIZATION

Daniel Kressner, Alice Cortinovis

This work is concerned with the computation of the action of a matrix function f(A), such as the matrix exponential or the matrix square root, on a vector b. For a general matrix A, this can be done by computing the compression of A onto a suitable Krylov ...

Siam Publications2024