Mixing
Explicit formula for inner product given orthonormal basis
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Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?
linear-algebra complex-numbers algorithms inner-product-space orthonormal
asked Jan 25 at 14:05
enedilenedil
1,524620
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add a comment |
$begingroup$
Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?
linear-algebra complex-numbers algorithms inner-product-space orthonormal
asked Jan 25 at 14:05
enedilenedil
1,524620
$endgroup$
add a comment |
$begingroup$
Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?
linear-algebra complex-numbers algorithms inner-product-space orthonormal
asked Jan 25 at 14:05
enedilenedil
1,524620
$endgroup$
Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?
linear-algebra complex-numbers algorithms inner-product-space orthonormal
linear-algebra complex-numbers algorithms inner-product-space orthonormal
asked Jan 25 at 14:05
enedilenedil
1,524620
asked Jan 25 at 14:05
enedilenedil
1,524620
asked Jan 25 at 14:05
enedilenedil
1,524620
asked Jan 25 at 14:05
enedilenedil
1,524620
asked Jan 25 at 14:05
enedilenedil
1,524620
1,524620
add a comment |
add a comment |
1 Answer
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$begingroup$
Using the sesquilinearity of the inner product, we have
$$
left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
$$
That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
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add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Using the sesquilinearity of the inner product, we have
$$
left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
$$
That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
add a comment |
$begingroup$
Using the sesquilinearity of the inner product, we have
$$
left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
$$
That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
add a comment |
$begingroup$
Using the sesquilinearity of the inner product, we have
$$
left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
$$
That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
$endgroup$
Using the sesquilinearity of the inner product, we have
$$
left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
$$
That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
answered Jan 25 at 14:11
OmnomnomnomOmnomnomnom
129k792185
129k792185
add a comment |
add a comment |
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