Is this projection matrix?

Hint:

You need to prove that $$\left((I-A)^T\right)^T = (I-A)^T, \quad \left((I-A)^T\right)^2 = (I-A)^T.$$


I don't understand what you mean by "we can see that $ I-A=I-A^2 => 0=A-A^2$ $A^2=A$ then $A-A=0$".

However

$$\begin{aligned} \left(I - A^T\right)^2 & = (I-A)^T(I-A)^T\\ &=\left((I-A)(I-A)\right)^T\\ &= (I-2A + A^2)^T\\ &=(I-A)^T \end{aligned}$$

So $(I-A)^T$ which satisfies the two required identities to be a projection is indeed a projection.

Tags:

Linear Algebra

Projection Matrices

Related

Finding the limit of $a_n = \frac{n+1}{2^{n+1}}\left(\frac{2}{1}+\frac{2^2}{2}+...+\frac{2^{n}}{n}\right)$ If $f(x+1/n)$ converges to a continuous function $f(x)$ uniformly on $\mathbb{R}$, is $f(x)$ necessarily uniformly continuous? Can the map sending a presentation to its group be considered as a functor? What is the smallest group containing all finite cyclic groups? Is there a graphical proof that $9n+1$ is triangular if $n$ is? Proof: not a perfect square If polynomial $P(x)$ has integer coefficients and at least three integer roots, then $P(x)+5^m$ has no more than one integer root for $m\geq 1$ Values of $a$ such that $x^5-x-a$ has quadratic factor Inequality $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}-\frac{c}{a+b}-\frac{a}{b+c}-\frac{b}{a+c}\ge 3/2$ What is a condition for two real functions $f,g$ to "commute", so $f(g(x))=g(f(x))$? Limit of $\frac{1}{M} \int_1^M M^{\frac{1}{x}} \textrm{d} x$ when $M \to +\infty$ Finding the number of real zeros for the equation $11^x + 13^x + 17^x-19^x=0 $

Recent Posts