Question

Is a matrix multiplied by itself always positive?I mean multiplied by its transposeie ATA >= 0, ?

Answer 1

Given a matrix A of dimension (m × n), where m is the number of rows and n is the number of columns.

The transpose of matrix A is obtained by interchanging the rows and columns.

This implies that

`\begin{gathered} \text{transpose of matrix A,} \\ A^T\text{ = (n }*\text{ m)} \\ \text{where } \\ n\text{ is the number of rows} \\ m\text{ is the number of columns} \end{gathered}`

For two matrices to be conformable for multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.

Thus, matrix A and its transpose are conformable since the number of columns in matrix A is equal to the number of rows in its transpose.

`\begin{gathered} A* A^T \\ (m* n)*(n* m) \end{gathered}`

When the matrix A and its transpose are multiplied, we obtain a square and symmetric matrix.