How to find a matrix closest to a given matrix in a Inner product space?
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Consider $M_2(mathbb{C})$ with the inner product $$langle A, B rangle = trace(B^*A)$$ where $*$ is conjugate transpose.
Find the closest element of the complex symmetric $2times 2$ matrices to $$A = begin{bmatrix}1&-i\ i&1end{bmatrix}$$How to approach this problem?
matrices inner-product-space least-squares
asked Nov 24 at 7:35
Mittal G
1,192515
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up vote
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down vote
favorite
Consider $M_2(mathbb{C})$ with the inner product $$langle A, B rangle = trace(B^*A)$$ where $*$ is conjugate transpose.
Find the closest element of the complex symmetric $2times 2$ matrices to $$A = begin{bmatrix}1&-i\ i&1end{bmatrix}$$How to approach this problem?
matrices inner-product-space least-squares
asked Nov 24 at 7:35
Mittal G
1,192515
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider $M_2(mathbb{C})$ with the inner product $$langle A, B rangle = trace(B^*A)$$ where $*$ is conjugate transpose.
Find the closest element of the complex symmetric $2times 2$ matrices to $$A = begin{bmatrix}1&-i\ i&1end{bmatrix}$$How to approach this problem?
matrices inner-product-space least-squares
asked Nov 24 at 7:35
Mittal G
1,192515
Consider $M_2(mathbb{C})$ with the inner product $$langle A, B rangle = trace(B^*A)$$ where $*$ is conjugate transpose.
Find the closest element of the complex symmetric $2times 2$ matrices to $$A = begin{bmatrix}1&-i\ i&1end{bmatrix}$$How to approach this problem?
matrices inner-product-space least-squares
matrices inner-product-space least-squares
asked Nov 24 at 7:35
Mittal G
1,192515
asked Nov 24 at 7:35
Mittal G
1,192515
asked Nov 24 at 7:35
Mittal G
1,192515
asked Nov 24 at 7:35
Mittal G
1,192515
asked Nov 24 at 7:35
Mittal G
1,192515
1,192515
add a comment |
add a comment |
1 Answer
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- Pick a basis $(e_1,e_2,e_3)$ of the space of all $2times2$ matrices.
- Use Gram-Schmidt to create from it an orthnormal basis $(f_1,f_2,f_3)$.
- The answer to your problem will be $langle A,f_1rangle f_1+langle A,f_2rangle f_2+langle A,f_3rangle f_3$
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
- Pick a basis $(e_1,e_2,e_3)$ of the space of all $2times2$ matrices.
- Use Gram-Schmidt to create from it an orthnormal basis $(f_1,f_2,f_3)$.
- The answer to your problem will be $langle A,f_1rangle f_1+langle A,f_2rangle f_2+langle A,f_3rangle f_3$
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
add a comment |
up vote
1
down vote
accepted
- Pick a basis $(e_1,e_2,e_3)$ of the space of all $2times2$ matrices.
- Use Gram-Schmidt to create from it an orthnormal basis $(f_1,f_2,f_3)$.
- The answer to your problem will be $langle A,f_1rangle f_1+langle A,f_2rangle f_2+langle A,f_3rangle f_3$
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
- Pick a basis $(e_1,e_2,e_3)$ of the space of all $2times2$ matrices.
- Use Gram-Schmidt to create from it an orthnormal basis $(f_1,f_2,f_3)$.
- The answer to your problem will be $langle A,f_1rangle f_1+langle A,f_2rangle f_2+langle A,f_3rangle f_3$
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
- Pick a basis $(e_1,e_2,e_3)$ of the space of all $2times2$ matrices.
- Use Gram-Schmidt to create from it an orthnormal basis $(f_1,f_2,f_3)$.
- The answer to your problem will be $langle A,f_1rangle f_1+langle A,f_2rangle f_2+langle A,f_3rangle f_3$
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
answered Nov 24 at 7:49
José Carlos Santos
147k22117218
147k22117218
add a comment |
add a comment |
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