Projection of matrix in a vector direction
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I have a question on matrix projection.
For any matrix : $S$
$$S = sum_i lambda_i v_i v_i^T$$
So to have a component of this matrix in arbitrary $u_r$ and $u_o$ directions. Here ${u_r, u_o}$ are arbitrary orthonormal basis.
I can write the projection into one direction as,
$$lambda_r = u_r^T S u_r = sum_i lambda_i langle v_i cdot u_r rangle^2 $$
but, I can also write the orthonormal decomposition of eigenvectors of $S$ in basis ${u_r, u_o}$, which would give me,
$$lambda_r = sum_i langle lambda_i v_i cdot u_r rangle = sum_i lambda_i langle v_i cdot u_r rangle $$
Why is there this discrepancy in the two definition for what I think is the same quantity?
linear-algebra projection
asked Nov 13 at 10:07
hadi k
1263
add a comment |
up vote
0
down vote
favorite
I have a question on matrix projection.
For any matrix : $S$
$$S = sum_i lambda_i v_i v_i^T$$
So to have a component of this matrix in arbitrary $u_r$ and $u_o$ directions. Here ${u_r, u_o}$ are arbitrary orthonormal basis.
I can write the projection into one direction as,
$$lambda_r = u_r^T S u_r = sum_i lambda_i langle v_i cdot u_r rangle^2 $$
but, I can also write the orthonormal decomposition of eigenvectors of $S$ in basis ${u_r, u_o}$, which would give me,
$$lambda_r = sum_i langle lambda_i v_i cdot u_r rangle = sum_i lambda_i langle v_i cdot u_r rangle $$
Why is there this discrepancy in the two definition for what I think is the same quantity?
linear-algebra projection
asked Nov 13 at 10:07
hadi k
1263
I dont understand, you are projecting a matrix in the direction of a vector?
– Javi
Nov 15 at 3:44
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a question on matrix projection.
For any matrix : $S$
$$S = sum_i lambda_i v_i v_i^T$$
So to have a component of this matrix in arbitrary $u_r$ and $u_o$ directions. Here ${u_r, u_o}$ are arbitrary orthonormal basis.
I can write the projection into one direction as,
$$lambda_r = u_r^T S u_r = sum_i lambda_i langle v_i cdot u_r rangle^2 $$
but, I can also write the orthonormal decomposition of eigenvectors of $S$ in basis ${u_r, u_o}$, which would give me,
$$lambda_r = sum_i langle lambda_i v_i cdot u_r rangle = sum_i lambda_i langle v_i cdot u_r rangle $$
Why is there this discrepancy in the two definition for what I think is the same quantity?
linear-algebra projection
asked Nov 13 at 10:07
hadi k
1263
I have a question on matrix projection.
For any matrix : $S$
$$S = sum_i lambda_i v_i v_i^T$$
So to have a component of this matrix in arbitrary $u_r$ and $u_o$ directions. Here ${u_r, u_o}$ are arbitrary orthonormal basis.
I can write the projection into one direction as,
$$lambda_r = u_r^T S u_r = sum_i lambda_i langle v_i cdot u_r rangle^2 $$
but, I can also write the orthonormal decomposition of eigenvectors of $S$ in basis ${u_r, u_o}$, which would give me,
$$lambda_r = sum_i langle lambda_i v_i cdot u_r rangle = sum_i lambda_i langle v_i cdot u_r rangle $$
Why is there this discrepancy in the two definition for what I think is the same quantity?
linear-algebra projection
linear-algebra projection
asked Nov 13 at 10:07
hadi k
1263
asked Nov 13 at 10:07
hadi k
1263
asked Nov 13 at 10:07
hadi k
1263
asked Nov 13 at 10:07
hadi k
1263
asked Nov 13 at 10:07
hadi k
1263
1263
I dont understand, you are projecting a matrix in the direction of a vector?
– Javi
Nov 15 at 3:44
add a comment |
I dont understand, you are projecting a matrix in the direction of a vector?
– Javi
Nov 15 at 3:44
I dont understand, you are projecting a matrix in the direction of a vector?
– Javi
Nov 15 at 3:44
I dont understand, you are projecting a matrix in the direction of a vector?
– Javi
Nov 15 at 3:44
add a comment |
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I dont understand, you are projecting a matrix in the direction of a vector?
– Javi
Nov 15 at 3:44