Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\mathbf{v}$ be a non-zero (column) vector in $\mathbb{R}^n$.

(a) Find an explicit formula for the matrix $P_\mathbf{v}$ corresponding to the projection of $\mathbb{R}^n$ to the orthogonal complement of the one-dimensional subspace spanned by $\mathbf{v}$.

(b) What are the eigenvalues and eigenvectors of $P_\mathbf{v}$? Compute the dimensions of the associated eigenspaces. Justify your answers.

Wouldn't the matrix be a diagonal matrix such that the eigenvalues are $0$ and $1$ or $-1$? Since it is an orthogonal complement of the subspace spanned by $\mathbf{v}$, wouldn't the matrix also have rows that contain the null space?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

The orthogonal projection onto the one-dimensional subspace spanned by $v$ is $\frac{1}{\| v \|^2}v v^{\top}$. The orthogonal projection onto the complement of $\mathbb{R}v$ is $Id-\frac{1}{\| v \|^2}v v^{\top}$.

The eigenvectors of $Id-\frac{1}{\| v \|^2}v v^{\top}$ are: the vectors of $\mathbb{R}v$ with eigenvalue $0$, and the vectors of the complement of $\mathbb{R}v$ with eigenvalue $1$.

The dimension of $\mathbb{R}v$ is $1$, as $v \neq 0$. The dimension of the complement is $n-1$.

share|improve this answer
    
Thank you very much, this is a very concise answer! –  tamefoxes Oct 8 '12 at 15:54

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.