Projection question with eigenvalues and eigenvectors

In summary, the conversation discusses finding an explicit formula for the matrix Pv that corresponds to the projection of Rn to the orthogonal complement of a one-dimensional subspace spanned by a non-zero vector v. It also explores the eigenvalues and eigenvectors of Pv, concluding that the eigenvalues would be 0, 1, or -1 and the associated eigenspaces would all be (n-1) dimensional.
  • #1
leej72
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0

Homework Statement



Let v be a non-zero (column) vector in Rn.

(a) Find an explicit formula for the matrix Pv corresponding to the projection of Rn to the orthogonal complement of the one-dimensional subspace spanned by v.

(b) What are the eigenvalues and eigenvectors of Pv? Compute the dimensions of the associated eigenspaces. Justify your answers


The Attempt at a Solution



Wouldn't the formula for the matrix Pv be the null space for v such that it has eigenvalues 0,1 or -1?
 
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  • #2
Since the projection of Rn to the orthogonal complement of the one-dimensional subspace spanned by v would be a projection onto an (n-1) dimensional space, wouldn't the eigenvalues all be 0 and the eigenspaces all be (n-1) dimensional?
 

Related to Projection question with eigenvalues and eigenvectors

1. What is a projection question with eigenvalues and eigenvectors?

A projection question with eigenvalues and eigenvectors is a type of mathematical problem that involves finding the projections of a vector onto a subspace or a particular direction. This is done by using the eigenvalues and eigenvectors of a matrix.

2. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra. Eigenvalues represent the scalar values that when multiplied by an eigenvector, produce the same vector as the result. Eigenvectors are the corresponding vectors that do not change direction when multiplied by the eigenvalue.

3. How are eigenvalues and eigenvectors used in projection questions?

In projection questions, eigenvalues and eigenvectors are used to find the projections of a vector onto a subspace or a particular direction. This is done by first finding the eigenvalues and eigenvectors of a matrix, and then using them to construct a projection matrix that can be used to project the vector onto the desired direction.

4. What are some real-world applications of projection questions with eigenvalues and eigenvectors?

Projection questions with eigenvalues and eigenvectors are commonly used in computer graphics, where they are used to rotate and scale objects. They are also used in image and signal processing, as well as in machine learning algorithms, such as principal component analysis.

5. Are there any limitations to using eigenvalues and eigenvectors in projection questions?

One limitation of using eigenvalues and eigenvectors in projection questions is that they only work for square matrices. Additionally, they may not always provide an accurate solution, as the eigenvectors may not be unique or the matrix may not have distinct eigenvalues. In these cases, other methods, such as the Singular Value Decomposition (SVD), may be used for projection.

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