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arthurhenry
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Why is the Trace of a projection is its Rank.
Thank you
Thank you
A projection matrix is a square matrix that results in the same matrix when multiplied by itself. It represents a linear transformation that projects vectors onto a lower dimensional subspace.
The rank of a projection matrix is defined as the number of linearly independent rows or columns in the matrix. This is equal to the number of non-zero eigenvalues of the matrix.
The trace of a matrix is defined as the sum of its diagonal elements. For a projection matrix, the diagonal elements are all either 0 or 1. Thus, the trace of a projection matrix is equal to the number of 1's on its diagonal, which is also the rank of the matrix.
The rank of a projection matrix is equal to the number of dimensions in its image space, which is the complement of its null space. This means that the dimension of the null space is equal to the difference between the dimension of the input space and the rank of the projection matrix.
No, the rank of a projection matrix can never be greater than its dimension. This is because the rank is equal to the number of linearly independent rows or columns, and a square matrix cannot have more linearly independent rows or columns than its dimension.