Let A be a square matrix. A square root of A is a matrix B of...

 

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Let A be a square matrix. A square root of A is a matrix B of the same order such that B=A How to find a square root for a given matrix? If the matrix is symmetric, positive definite, there is a way to do that: (1) First find the Schur decomposition of A=QDQ! (2) Since the matrix is positive definite, the diagonal entries of D are non-negative. So, by taking square roots of these diagonal entries, one obtains a diagonal matrix S such that S=D. (3) Define then B=QSQ. Then B is also symmetric positive definite (why?), and B=A (why?) Let's experiment: Let A be the matrix 290 34 34 290 and let B be its square root. Then B(1,1)= and B(1,2)= Hint: You may surely find the Schur decomposition of A by hand, by finding the e- values and the e-vectors of it. But the command [Evec, Eval]-eig(A) finds those too. Let A be a square matrix. A square root of A is a matrix B of the same order such that B=A How to find a square root for a given matrix? If the matrix is symmetric, positive definite, there is a way to do that: (1) First find the Schur decomposition of A=QDQ! (2) Since the matrix is positive definite, the diagonal entries of D are non-negative. So, by taking square roots of these diagonal entries, one obtains a diagonal matrix S such that S=D. (3) Define then B=QSQ. Then B is also symmetric positive definite (why?), and B=A (why?) Let's experiment: Let A be the matrix 290 34 34 290 and let B be its square root. Then B(1,1)= and B(1,2)= Hint: You may surely find the Schur decomposition of A by hand, by finding the e- values and the e-vectors of it. But the command [Evec, Eval]-eig(A) finds those too.