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Real Matrix-Matrix Products
Inverse Elements
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Affine MMSE Estimators
Least Squares Linear Regressors
Matrix Determinant of Inverses
Matrix Inverses
Matrix Product Inverses
Matrix Similarity
Monotonic Functions of Real Matrices
Multivariate Normals
Permutation Matrices
Probabilistic Errors Linear Model
Real General Linear Groups
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Real Matrix Inverses


Let $A \in \R ^{m \times n}$ and define $f: \R ^n \to \R ^m$ by $f(x) = Ax$. Then $f$ is a linear function from $\R ^{n}$ to $\R ^{m}$. Conversely, suppose $g: \R ^n \to \R ^m$ is a linear function. Then there exists a matrix $B \in \R ^{m \times n}$ so that $g(z) = Bz$. Does this function have an inverse?


If $A \in \R ^{m \times n}$, with $m \neq n$, then the inverse of $f$ can not exist. For a square matrix $A \in \R ^{n \times n}$, $B \in \R ^{n \times n}$ is a left inverse if $BA = I$. In other words, $B$ is a left inverse element of $A$ in the algebra of matrices with the operation of multiplication. $C \in \R ^{n \times n}$ is a right inverse if $AC = I$.


We call a square matrix $A$ invertible if there is $B \in \R ^{n \times n}$ so that $BA = I$.

Now suppose that $A \in \R ^{n \times n}$. Of course, the inverse may not exist. Consider, for example if $A$ was the $n$ by $n$ matrix of zeros. If there exists a matrix $B$ so that $BA = I$ we call $B$ the left inverse of $A$ and likewise if $AC = I$ we call $C$ the right inverse of $A$. In the case that $A$ is square, the right inverse and left inverse coincide.

Suppose $A, B, C \in \R ^{n \times n}$. If $BA = I$ and $AC = I$, then $B = C$.
Since $BA = AC$ we have $BBA = BAC$ so $B = C$ since $BA = I$.
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