# A real-valued$\Rightarrow$Matrix $S$ of Cholesky-decomposition $A=SS^T$ is real valued.

Let $$A$$ be a symmetric positive definite invertible real valued matrix. Then we can write $$A=SS^T$$ where $$S$$ is a lower triangular matrix with positive entries on its diagonal. This decomposition is called the Cholesky decomposition.

How do we know that the entries of $$S$$ are themselves real-valued? I need this property in order to solve a problem related to the Courant-Fischer min-max theorem.

• you should add the symmetric condition – Ahmad Bazzi Dec 13 '19 at 11:30
• $S$ can be computed by forward substitution. The substitution process does not require field extension. Therefore $S$ is real. – user1551 Dec 13 '19 at 13:52
• What is "forward substitution" ? – Christian Singer Dec 13 '19 at 14:43
• "Forward substitution" refers to the practice of solving an indexed set of equations iteratively, in which we substitute the solution to the $(k-1)$-th equation into the $k$-th equation and solve the latter. You may think of it as mathematical induction. – user1551 Dec 14 '19 at 11:12

Let $$A_k$$ be the leading principal $$k\times k$$ submatrix of $$A$$. Clearly, $$A_1=S_1S_1^T$$ where $$S_1=\sqrt{A_1}$$ is a real $$1\times1$$ lower triangular matrix. Now suppose that for some $$k$$, $$A_{k-1}=S_{k-1}S_{k-1}^T$$ for some real lower triangular matrix $$S_{k-1}$$. Since $$A_{k-1}$$ is positive definite, $$S_{k-1}$$ is nonsingular. Also, as $$u^TA_ku>0$$ for every nonzero vector $$u$$, if we write $$A_k=\pmatrix{A_{k-1}&v_k\\ v_k^T&a_k}$$ and put $$u=(-v_k^TA_{k-1}^{-1},\,1)^T$$, we obtain $$a_k-v_k^TA_{k-1}^{-1}v_k>0$$. Therefore the equation $$\pmatrix{A_{k-1}&v_k\\ v_k^T&a_k} =\pmatrix{S_{k-1}&0\\ x^T&s}\pmatrix{S_{k-1}^T&x\\ 0&s}\tag{1}$$ has a unique solution $$x=S_{k-1}^{-1}v_k,\,s=\sqrt{a_k-x^Tx}=\sqrt{a_k-v_k^TA_{k-1}^{-1}v_k}$$. This means $$A_k=S_kS_k^T$$ for some real lower triangular matrix $$S_k$$. So, if we start from $$A_1$$ and keep solving $$(1)$$ for $$k=2,3,\ldots$$, we see that $$A=SS^T$$ for some real lower triangular matrix $$S$$.