transpose of positive matrix is positive
$begingroup$
how to prove it?
I am talking about matrixes which satisfy:
$$( Ax , x ) > 0quad text{ for any}quad ;x neq 0.$$
How to prove that $A^T;$ is also positive?
$$x^T A x = ( x^T A x )^T$$
and what?
linear-algebra matrices
edited Jan 1 at 21:45
user376343
3,9584829
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
$endgroup$
add a comment |
$begingroup$
how to prove it?
I am talking about matrixes which satisfy:
$$( Ax , x ) > 0quad text{ for any}quad ;x neq 0.$$
How to prove that $A^T;$ is also positive?
$$x^T A x = ( x^T A x )^T$$
and what?
linear-algebra matrices
edited Jan 1 at 21:45
user376343
3,9584829
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
$endgroup$
add a comment |
$begingroup$
how to prove it?
I am talking about matrixes which satisfy:
$$( Ax , x ) > 0quad text{ for any}quad ;x neq 0.$$
How to prove that $A^T;$ is also positive?
$$x^T A x = ( x^T A x )^T$$
and what?
linear-algebra matrices
edited Jan 1 at 21:45
user376343
3,9584829
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
$endgroup$
how to prove it?
I am talking about matrixes which satisfy:
$$( Ax , x ) > 0quad text{ for any}quad ;x neq 0.$$
How to prove that $A^T;$ is also positive?
$$x^T A x = ( x^T A x )^T$$
and what?
linear-algebra matrices
linear-algebra matrices
edited Jan 1 at 21:45
user376343
3,9584829
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
edited Jan 1 at 21:45
user376343
3,9584829
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
edited Jan 1 at 21:45
user376343
3,9584829
edited Jan 1 at 21:45
user376343
3,9584829
edited Jan 1 at 21:45
user376343
3,9584829
3,9584829
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
asked Jan 17 '11 at 8:53
mr. Vachovskymr. Vachovsky
1214
1214
add a comment |
add a comment |
1 Answer
1
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votes
$begingroup$
Hint: Write the inner product as $x^T A x$ and use the fact that that expression is its own transpose (since it's a 1-by-1 matrix).
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
$endgroup$
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Hint: Write the inner product as $x^T A x$ and use the fact that that expression is its own transpose (since it's a 1-by-1 matrix).
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
$endgroup$
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
add a comment |
$begingroup$
Hint: Write the inner product as $x^T A x$ and use the fact that that expression is its own transpose (since it's a 1-by-1 matrix).
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
$endgroup$
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
add a comment |
$begingroup$
Hint: Write the inner product as $x^T A x$ and use the fact that that expression is its own transpose (since it's a 1-by-1 matrix).
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
$endgroup$
Hint: Write the inner product as $x^T A x$ and use the fact that that expression is its own transpose (since it's a 1-by-1 matrix).
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
answered Jan 17 '11 at 8:55
Hans LundmarkHans Lundmark
36k564115
36k564115
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
add a comment |
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
$begingroup$
Thx, I understood!
$endgroup$
– mr. Vachovsky
Jan 17 '11 at 9:08
add a comment |
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