Is matrix $A$ unique?

Algebra Level pending

Consider two symmetric $n \times n$ positive definite matrices $Q_1$ and $Q_2$ , where $n \ge 2$ , we want to find a matrix $A$ such that

$A^T Q_1 A = Q_2$

What can be said about matrix $A$ ?

Matrix

A

does not necessarily exist There is an infinite number of matrices

A

There is a finite number greater than two of matrices

A

Matrix

A

is unique There is exactly two such matrices

A

1 solution

Tom Engelsman
May 8, 2021

For simplicity, let's focus on the case $n=2$ with:

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, Q_{1} = \begin{bmatrix} x & y \\ y & z \end{bmatrix} (x > 0, xz-y^2 > 0$ for positive-definiteness)

and $Q_{2} = A^{T}Q_{1} A = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \begin{bmatrix} x & y \\ y & z \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a(ax+cy)+c(ay+cz) & a(bx+dy)+c(by+dz) \\ b(ax+cy)+d(ay+cz) & b(bx+dy)+d(by+dz) \end{bmatrix}$ .

In order for $Q_{2} = Q^{T}_{2}$ , we require $Q_{2(1,2)} = Q_{2(2,1)} \Rightarrow a(bx+dy)+c(by+dz) = b(ax+cy)+d(ay+cz)$ ;

or $(abx-bax) + (ady - day) + (cby - bcy) + (cdz-dcz) = 0$ ;

or $0 = 0$ for all $a,b,c,d$ .

which means there exists an infinite number of matrices $A$ .

Is matrix A A A unique?

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Is matrix $A$ unique?