An orthogonal projection matrix is a square matrix that projects a vector onto a subspace, while preserving its length and perpendicularity to the subspace. It is also known as an orthogonal projection operator.
To prove that a matrix is idempotent, you need to show that when the matrix is multiplied by itself, it results in the same matrix. In the case of an orthogonal projection matrix, this can be shown by multiplying the matrix by its transpose, which will result in the same matrix.
An idempotent orthogonal projection matrix ensures that the projection operation can be applied multiple times without changing the result. This is useful in many applications, such as in signal processing and image compression.
No, an orthogonal projection matrix must be idempotent by definition. If the matrix is not idempotent, it means that the projection operation is not preserving the length and perpendicularity of the vector, which goes against the properties of an orthogonal projection matrix.
An orthogonal projection matrix is a special case of a regular projection matrix, where the subspace onto which the vector is projected is orthogonal to the complement of the subspace. This ensures that the projection is done in a way that preserves the length and perpendicularity of the vector, while a regular projection matrix may not have these properties.