Solving a Puzzling Problem: No A Exists for Symmetric Matrix B

[BOS200011]

Give an example of a 2X2 symmetric matrix B that cannot be written as B = A^TA. Give an explanation as to why no such A exists for the matrix B you have given.


I know that the product A^TA is a symmetric matrix, but how could there be no such A that exists for some matrix B?

I'm really stuck on this problem, and I would appreciate it if anyone could help. Thank you so much in advance.

 

[D H]

What is the nature of the eigenvalues of A^TA?

 

[Fredrik]

If you write

$$A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$

and calculate A^TA, it's immediately obvious that the result is not the most general form of a symmetric 2×2 matrix. No more calculations are necessary. Just stare at the result until you get it.

 

[BOS200011]

1 2
2 3

I did out the product of A^TA and came up with this matrix B that couldn't be written as such. Am I correct?

 

[acarchau]

You are correct about the above matrix. Try to figure out why (hint: determinants).
What do you think about the following matrix:
$$A=\begin{pmatrix}1&10\\ 10&-1\end{pmatrix}$$

 

[D H]

Determinants are quite the right answer, either. Both the identity matrix and its additive inverse,

$$\bmatrix -1 & 0 \\ 0 & -1\endbmatrix$$

have the same determinant. The identity matrix can obviously be written in the form A^TA. Its additive inverse cannot.

 

[Fredrik]

The OP's task was just to find an example of a symmetric matrix that can't be expressed as A^TA, and noting that det A^TA≥0 is certainly a good start. It explains why

1 2
2 3

is the kind of matrix we're looking for. My idea was to note that the elements on the diagonal of A^TA are ≥0. That explains both

1 10
10 -1

and

-1 0
0 -1

Not sure if there are symmetric matrices with both the determinant and the diagonal elements ≥0 that can't be written as A^TA. I'm too tired to think about that right now.

 

[D H]

You can get away with looking at the determinant and the diagonal elements for a 2x2. That trick won't work for anything larger than 2x2. What always works is to look at the eigenvalues, like I said in post #2. Given a real nxm matrix A, the eigenvalues of the matrix A^TA are always non-negative. If any eigenvalue of some matrix B is negative then it is impossible to write B in the form A^TA.