If $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix?

Problem 426

A square matrix $A$ is called idempotent if $A^2=A$.

(a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix.

(b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then determine all integers $k$ such that the matrix $I-kA$ is idempotent.

(c) Let $A$ and $B$ be $n\times n$ matrices satisfying
$AB=A \text{ and } BA=B.$ Then prove that $A$ is an idempotent matrix.

Proof.

(a) Prove that the matrix $I-A$ is an idempotent matrix.

To prove that $I-A$ is an idempotent matrix, we show that $(I-A)^2=I-A$.
We compute
\begin{align*}
(I-A)^2&=(I-A)(I-A)\\
&=I(I-A)-A(I-A)\\
&=I-A-A+A^2\\
&=I-2A+A^2\\
&=I-2A+A && \text{since $A$ is idempotent}\\
&=I-A.
\end{align*}
Thus, we have $(I-A)^2=I-A$ and $I-A$ is an idempotent matrix.

(b) Determine all integers $k$ such that the matrix $I-kA$ is idempotent.

Let us find out the condition on $k$ so that $I-kA$ is an idempotent matrix.
We have
\begin{align*}
&(I-kA)^2=(I-kA)(I-kA)\\
&=I(I-kA)-kA(I-kA)\\
&=I-kA-kA+k^2A^2\\
&=I-2kA+k^2A && \text{ since $A$ is idempotent}\\
&=I-(2k-k^2)A.
\end{align*}

It follows that $I-kA$ is idempotent if and only if $I-kA=I-(2k-k^2)A$, or equivalently $(k^2-k)A=O$, the zero matrix.
Since $A$ is not the zero matrix, we see that $I-kI$ is idempotent if and only if $k^2-k=0$.

Since $k^2-k=k(k-1)$, we conclude that $I-kA$ is an idempotent matrix if and only if $k=0, 1$.

(c) Prove that $A$ is an idempotent matrix.

Let $A$ and $B$ be $n\times n$ matrices satisfying
$AB=A \tag{*}$ and
$BA=B. \tag{**}$ Our goal is to show that $A^2=A$.
We compute
\begin{align*}
&A^2=AA\\
&=(AB)A && \text{by (*)}\\
&=A(BA)\\
&=AB && \text{by (**)}\\
&=A && \text{by (*)}.
\end{align*}
Therefore, we have obtained the identity $A^2=A$, and we conclude that $A$ is an idempotent matrix.

Related Question.

Problem.
(a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.
Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.

Prove that $P$ is an idempotent matrix.

(b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.

Prove that $Q$ is an idempotent matrix.

(c) Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).

The proofs are given in the post ↴
Unit Vectors and Idempotent Matrices

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