For an orthogonal matrix $Q$, prove $operatornamecond(Q)=1$Orthogonal matrix normWhy is the matrix product of 2 orthogonal matrices also an orthogonal matrix?Finding an orthogonal Matrix with distinct eigenvaluesOrthogonal Matrix Eigenvaluehow to prove this matrix is orthogonalShow that for $n times n$ orthogonal matrix $A$ that $operatornameCond(A)leq n$Can we prove that the eigenvalues of an $ntimes n$ orthogonal matrix are $pm 1$ by only using the definition of orthogonal matrix?Show non-singularity of orthogonal matrixProve that the multiples of two orthogonal eigenvectors with a matrix are also orthogonalProve that $-1$ is an eigenvalue of an orthogonal matrix $A in M_4 times 4 (Bbb R)$ with $det(A)=-1$Band matrix conjugation relation between orthogonal matrices
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For an orthogonal matrix $Q$, prove $operatornamecond(Q)=1$
Orthogonal matrix normWhy is the matrix product of 2 orthogonal matrices also an orthogonal matrix?Finding an orthogonal Matrix with distinct eigenvaluesOrthogonal Matrix Eigenvaluehow to prove this matrix is orthogonalShow that for $n times n$ orthogonal matrix $A$ that $operatornameCond(A)leq n$Can we prove that the eigenvalues of an $ntimes n$ orthogonal matrix are $pm 1$ by only using the definition of orthogonal matrix?Show non-singularity of orthogonal matrixProve that the multiples of two orthogonal eigenvectors with a matrix are also orthogonalProve that $-1$ is an eigenvalue of an orthogonal matrix $A in M_4 times 4 (Bbb R)$ with $det(A)=-1$Band matrix conjugation relation between orthogonal matrices
$begingroup$
Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$
I succeed to solve it with eigenvalues but I'm looking for an easier way.
matrices orthogonal-matrices condition-number spectral-norm
edited yesterday
Rodrigo de Azevedo
13.1k41960
asked yesterday
J. DoeJ. Doe
16811
$endgroup$
add a comment |
$begingroup$
Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$
I succeed to solve it with eigenvalues but I'm looking for an easier way.
matrices orthogonal-matrices condition-number spectral-norm
edited yesterday
Rodrigo de Azevedo
13.1k41960
asked yesterday
J. DoeJ. Doe
16811
$endgroup$
1
$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday
add a comment |
$begingroup$
Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$
I succeed to solve it with eigenvalues but I'm looking for an easier way.
matrices orthogonal-matrices condition-number spectral-norm
edited yesterday
Rodrigo de Azevedo
13.1k41960
asked yesterday
J. DoeJ. Doe
16811
$endgroup$
Given an orthogonal matrix $Q$, prove $$|Q|_2cdot |Q^-1|_2=1$$
I succeed to solve it with eigenvalues but I'm looking for an easier way.
matrices orthogonal-matrices condition-number spectral-norm
matrices orthogonal-matrices condition-number spectral-norm
edited yesterday
Rodrigo de Azevedo
13.1k41960
asked yesterday
J. DoeJ. Doe
16811
edited yesterday
Rodrigo de Azevedo
13.1k41960
asked yesterday
J. DoeJ. Doe
16811
edited yesterday
Rodrigo de Azevedo
13.1k41960
edited yesterday
Rodrigo de Azevedo
13.1k41960
edited yesterday
Rodrigo de Azevedo
13.1k41960
13.1k41960
asked yesterday
J. DoeJ. Doe
16811
asked yesterday
J. DoeJ. Doe
16811
asked yesterday
J. DoeJ. Doe
16811
16811
1
$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday
add a comment |
1
$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday
1
1
$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday
$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.
So:
$$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
Since $Q^-1$ is orthogonal as well, the result follows.
answered yesterday
I like SerenaI like Serena
4,2721722
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1 Answer
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active
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oldest
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active
oldest
votes
$begingroup$
An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.
So:
$$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
Since $Q^-1$ is orthogonal as well, the result follows.
answered yesterday
I like SerenaI like Serena
4,2721722
$endgroup$
add a comment |
$begingroup$
An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.
So:
$$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
Since $Q^-1$ is orthogonal as well, the result follows.
answered yesterday
I like SerenaI like Serena
4,2721722
$endgroup$
add a comment |
$begingroup$
An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.
So:
$$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
Since $Q^-1$ is orthogonal as well, the result follows.
answered yesterday
I like SerenaI like Serena
4,2721722
$endgroup$
An orthogonal matrix preserves distance (it's an isometry). It means that for any $mathbf x$ we have $|Qmathbf x|_2=|mathbf x|_2$.
So:
$$|Q|_2 oversettextdef= sup_mathbf xne mathbf 0frac_2 = sup_mathbf xne mathbf 0frac_2_2=1$$
Since $Q^-1$ is orthogonal as well, the result follows.
answered yesterday
I like SerenaI like Serena
4,2721722
answered yesterday
I like SerenaI like Serena
4,2721722
answered yesterday
I like SerenaI like Serena
4,2721722
answered yesterday
I like SerenaI like Serena
4,2721722
4,2721722
add a comment |
add a comment |
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$begingroup$
An orthogonal matrix actually has operator norm equal to $1$. And if $Q$ is orthogonal, then so is $Q^-1$.
$endgroup$
– Minus One-Twelfth
yesterday