Friday, 1 April 2016

linear algebra - A inverse times A in the middle of a matrix multiplication



I was wondering if for the below matrix multiplication:



$A^T * A *\ A^{-1} * (A^{-1})^T$




we can assume the product of the inner 2 matrices to equal the identity matrix I, and simply rewrite this as:



$A^T * (A^{-1})^T$



or is this not generally acceptable because matrix multiplication is not commutative?


Answer



You are correct. Since matrix multiplication is associative, thus you can do



$A^T \cdot A \cdot A^{-1} \cdot(A^{-1})^T = A^T \cdot (A \cdot A^{-1}) \cdot (A^{-1})^T = A^T \cdot (A^{-1})^T$




But you can even go further by switching the inverse with the transpose:



$A^T \cdot (A^{-1})^T = A^T \cdot (A^{T})^{-1} = I $



So your whole expression is equal to the identity matrix.


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