Expression for $|Ax|_2^2$ in terms of rows of $A$Matrix, Ranks and RowsMatrix Multiplication By RowsRewrite an expression in terms of basis vectorsDeterminant when rows reversedReformulating a vector-matrix expression in terms of laplacianElimination and exchanging rowsgather terms of expressionWriting this matrix expression in terms of vec operatorSimilar concentration results to Restricted Isometry Property?Vandermonde matrices nullspaces and notation
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Expression for $|Ax|_2^2$ in terms of rows of $A$
Matrix, Ranks and RowsMatrix Multiplication By RowsRewrite an expression in terms of basis vectorsDeterminant when rows reversedReformulating a vector-matrix expression in terms of laplacianElimination and exchanging rowsgather terms of expressionWriting this matrix expression in terms of vec operatorSimilar concentration results to Restricted Isometry Property?Vandermonde matrices nullspaces and notation
$begingroup$
I am currently reading 1011.3027, Roman Vershynin's introduction to non-asymptotic random matrices. On page 24, there is an equation, (5.24), which reads
beginalign*
|Ax|_2^2 = sum_i=1^nlangle A_i,xrangle^2,
endalign*
where $A_iin mathbbR^n$ are the rows of the matrix $Ain mathcalM_rtimes n$ and $xin mathbbR^n$ is a vector (on the unit sphere).
I find myself believing this to be true, but I am not sure how to actually prove it. I tried playing around with an SVD and also decomposing $A$ as a sum of basic matrices but I didn't get anywhere useful.
Thus, I would appreciate a reference or proof of the above equation.
Thanks in advance.
linear-algebra matrices
edited yesterday
Rodrigo de Azevedo
13k41960
asked yesterday
afightingchanceafightingchance
516
$endgroup$
add a comment |
$begingroup$
I am currently reading 1011.3027, Roman Vershynin's introduction to non-asymptotic random matrices. On page 24, there is an equation, (5.24), which reads
beginalign*
|Ax|_2^2 = sum_i=1^nlangle A_i,xrangle^2,
endalign*
where $A_iin mathbbR^n$ are the rows of the matrix $Ain mathcalM_rtimes n$ and $xin mathbbR^n$ is a vector (on the unit sphere).
I find myself believing this to be true, but I am not sure how to actually prove it. I tried playing around with an SVD and also decomposing $A$ as a sum of basic matrices but I didn't get anywhere useful.
Thus, I would appreciate a reference or proof of the above equation.
Thanks in advance.
linear-algebra matrices
edited yesterday
Rodrigo de Azevedo
13k41960
asked yesterday
afightingchanceafightingchance
516
$endgroup$
add a comment |
$begingroup$
I am currently reading 1011.3027, Roman Vershynin's introduction to non-asymptotic random matrices. On page 24, there is an equation, (5.24), which reads
beginalign*
|Ax|_2^2 = sum_i=1^nlangle A_i,xrangle^2,
endalign*
where $A_iin mathbbR^n$ are the rows of the matrix $Ain mathcalM_rtimes n$ and $xin mathbbR^n$ is a vector (on the unit sphere).
I find myself believing this to be true, but I am not sure how to actually prove it. I tried playing around with an SVD and also decomposing $A$ as a sum of basic matrices but I didn't get anywhere useful.
Thus, I would appreciate a reference or proof of the above equation.
Thanks in advance.
linear-algebra matrices
edited yesterday
Rodrigo de Azevedo
13k41960
asked yesterday
afightingchanceafightingchance
516
$endgroup$
I am currently reading 1011.3027, Roman Vershynin's introduction to non-asymptotic random matrices. On page 24, there is an equation, (5.24), which reads
beginalign*
|Ax|_2^2 = sum_i=1^nlangle A_i,xrangle^2,
endalign*
where $A_iin mathbbR^n$ are the rows of the matrix $Ain mathcalM_rtimes n$ and $xin mathbbR^n$ is a vector (on the unit sphere).
I find myself believing this to be true, but I am not sure how to actually prove it. I tried playing around with an SVD and also decomposing $A$ as a sum of basic matrices but I didn't get anywhere useful.
Thus, I would appreciate a reference or proof of the above equation.
Thanks in advance.
linear-algebra matrices
linear-algebra matrices
edited yesterday
Rodrigo de Azevedo
13k41960
asked yesterday
afightingchanceafightingchance
516
edited yesterday
Rodrigo de Azevedo
13k41960
asked yesterday
afightingchanceafightingchance
516
edited yesterday
Rodrigo de Azevedo
13k41960
edited yesterday
Rodrigo de Azevedo
13k41960
edited yesterday
Rodrigo de Azevedo
13k41960
13k41960
asked yesterday
afightingchanceafightingchance
516
asked yesterday
afightingchanceafightingchance
516
asked yesterday
afightingchanceafightingchance
516
516
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: What is the $i$-th element of the vector $Ax$, using the definition of matrix multiplication?
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
$endgroup$
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
add a comment |
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1 Answer
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1 Answer
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active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Hint: What is the $i$-th element of the vector $Ax$, using the definition of matrix multiplication?
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
$endgroup$
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
add a comment |
$begingroup$
Hint: What is the $i$-th element of the vector $Ax$, using the definition of matrix multiplication?
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
$endgroup$
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
add a comment |
$begingroup$
Hint: What is the $i$-th element of the vector $Ax$, using the definition of matrix multiplication?
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
$endgroup$
Hint: What is the $i$-th element of the vector $Ax$, using the definition of matrix multiplication?
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
answered yesterday
Minus One-TwelfthMinus One-Twelfth
2,07219
2,07219
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
add a comment |
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
$begingroup$
Row-one-times-column-one the mantra goes... $(Ax)_i=langle A_i,xrangle$. Got it. Thanks!
$endgroup$
– afightingchance
yesterday
add a comment |
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