What is the meaning of subtracting from the identity matrix?

The difference $I - A$ is sometimes used with respect to projection matrices, where $A$ is the projection onto a subspace and $I - A$ is the projection onto the orthogonal complement of that subspace.


If there exists $n\in\mathbb{N}$ with $A^{n}=0$, then $I−A$ is invertible (try to prove this yourself before reading why!).

The reason: multiply out the brackets in the expression $$(I−A)(A^{n−1}+A^{n−2}+\cdots+A+I)$$ and see that you get $I$. It follows that $A^{n−1}+A^{n−2}+\cdots+A+I$ is the inverse of $I−A$ in this case.

By the way, a matrix $A$ with $A^{n}=0$ for some $n\in\mathbb{N}$ is called nilpotent.

By the way, the fact presented above can be used to give a proof of the (basic) fact that all eigenvalues of a nilpotent matrix must equal zero. In particular, this means that the determinant and trace of a nilpotent matrix are always $0$.

If $A$ is nilpotent then so is $\frac{1}{\lambda}A$ for any $\lambda\neq0$. Hence $I-\frac{1}{\lambda}A=\frac{1}{\lambda}(\lambda I-A)$ has an inverse. It follows that $\det{(\lambda I-A)}\neq0$ whenever $\lambda\neq0$ and $A$ is nilpotent, so the characteristic polynomial only has $0$ as a root.

So considering $I-A$ can actually come in handy to prove things.

Tags:

Linear Algebra

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