A property of positive definite matrices

Assume we have 2 positive definite matrices A and B . Show that there exists a non-singular matrix S such that –

SAS’ = I

SBS’ = L

Here I is the Identity matrix and L is a diagonal matrix. S’ is the transpose of S.

Solutions Collecting From Web of "A property of positive definite matrices"

Since $B$ is SPD, there is a $C$ such that $B=CC^T$. Consider the matrix $G=C^{-1}AC^{-T}$. Note that $G$ is SPD too. Let $G=UDU^T$ be the spectral decomposition of $G$ with $U$ orthogonal and $D$ diagonal. Then you can verify, that with $S=D^{1/2}U^TC^{-1}$, you get what you need.

This is Theorem 7.6.4 on Pg 465 in Horn & Johnson’s Matrix Analysis 1985. The proof is essentially what Algebraic Pavel suggested. Here is another dig at the same:

Since $B$ is positive-definite, $B = KK’$ for some invertible $K$ ($K’$ is the transpose of $K$).

Define $G = K^{-1}A(K^{-1})’$. We can see that $G$ is symmetric since

$$G’ = (K^{-1}A(K^{-1})’)’$$
$${}= ((K^{-1})’)'(K^{-1}A)’$$
$${}= K^{-1}A'(K^{-1})’$$
$${}= K^{-1}A(K^{-1})’$$
$${} = G$$

Thus, diagonalize $G$ into $UDU’$. Now define $S = D^{-1/2} U’ K^{-1}$. Then, $S’ = (K^{-1})’U D^{-1/2}$

Claim that $S$ satisfies the required conditions:

  • $SAS’ = D^{-1/2} U’ K^{-1} A (K^{-1})’U D^{-1/2} = D^{-1/2} U’ G U D^{-1/2} = D^{-1/2} D D^{-1/2} = I$

  • $SBS’ = D^{-1/2} U’ K^{-1} B (K^{-1})’U D^{-1/2} = D^{-1/2} U’ K^{-1} K K’ (K^{-1})’U D^{-1/2} = D^{-1/2} U’ U D^{-1/2} = D^{-1/2} D^{-1/2} = D^{-1}$. Define $L$ to be equal to $D^{-1}$.

Thus, we have constructed a non-singular $S$ such that $SAS’ = I$ and $SBS’ = L$ as needed.

Links to some related questions in SE:

  • Link1
  • Link2