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Linear and multilinear algebra; matrix theory Definition

Square root of a positive semidefinite matrix

Builds on Positive semidefinite matrix

Definition

Let AA be a positive semidefinite n×nn \times n complex Hermitian matrix. There is exactly one positive semidefinite matrix BB with B2=AB^2 = A. This matrix is the square root of AA, written A1/2A^{1/2} or A\sqrt{A}.

Remarks

If A=UDU∗A = U D U^* with UU unitary and DD diagonal with entries di≥0d_i \ge 0, then A1/2=UD1/2U∗A^{1/2} = U D^{1/2} U^*, with entries di\sqrt{d_i} in D1/2D^{1/2}. A1/2A^{1/2} commutes with each matrix that commutes with AA. Without the condition B⪰0B \succeq 0, a matrix can have many square roots; for example, I2I_2 has infinitely many.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Square_root_of_a_matrix, Positive semidefinite matrices
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