Relation between subspace distance and principal angles
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For two 1-dimensional (real) vector subspaces defined by the basis vectors $w_1$ and $w_2$ respectively, Golub and van Loan (Matrix Computations 4th ed. p. 331) state that the subspace distance is related to the principal angle -- which is actually just the inner product in this case -- as follows:
$$text{dist}(w_1,w_2) = ||w_1w_1^T - w_2w_2^T||_2 = sqrt{1-(w_1^Tw_2)^2}. $$
How do we go about showing this is true?
linear-algebra numerical-linear-algebra
asked Jan 23 at 16:27
HCNHCN
11
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add a comment |
$begingroup$
For two 1-dimensional (real) vector subspaces defined by the basis vectors $w_1$ and $w_2$ respectively, Golub and van Loan (Matrix Computations 4th ed. p. 331) state that the subspace distance is related to the principal angle -- which is actually just the inner product in this case -- as follows:
$$text{dist}(w_1,w_2) = ||w_1w_1^T - w_2w_2^T||_2 = sqrt{1-(w_1^Tw_2)^2}. $$
How do we go about showing this is true?
linear-algebra numerical-linear-algebra
asked Jan 23 at 16:27
HCNHCN
11
$endgroup$
add a comment |
$begingroup$
For two 1-dimensional (real) vector subspaces defined by the basis vectors $w_1$ and $w_2$ respectively, Golub and van Loan (Matrix Computations 4th ed. p. 331) state that the subspace distance is related to the principal angle -- which is actually just the inner product in this case -- as follows:
$$text{dist}(w_1,w_2) = ||w_1w_1^T - w_2w_2^T||_2 = sqrt{1-(w_1^Tw_2)^2}. $$
How do we go about showing this is true?
linear-algebra numerical-linear-algebra
asked Jan 23 at 16:27
HCNHCN
11
$endgroup$
For two 1-dimensional (real) vector subspaces defined by the basis vectors $w_1$ and $w_2$ respectively, Golub and van Loan (Matrix Computations 4th ed. p. 331) state that the subspace distance is related to the principal angle -- which is actually just the inner product in this case -- as follows:
$$text{dist}(w_1,w_2) = ||w_1w_1^T - w_2w_2^T||_2 = sqrt{1-(w_1^Tw_2)^2}. $$
How do we go about showing this is true?
linear-algebra numerical-linear-algebra
linear-algebra numerical-linear-algebra
asked Jan 23 at 16:27
HCNHCN
11
asked Jan 23 at 16:27
HCNHCN
11
asked Jan 23 at 16:27
HCNHCN
11
asked Jan 23 at 16:27
HCNHCN
11
asked Jan 23 at 16:27
HCNHCN
11
11
add a comment |
add a comment |
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