Mixing
Definiteness of matrix after Woodbury inversion.
$begingroup$
Consider a real, symmetric and positive definite $ntimes n$ matrix $mathbf{K}$, and a $ntimes m$ matrix $mathbf{W}$. $mathbf{W}$ contains $m$ columns with all zeros except a single entry in each column which is 1. Furthermore $(gmathbf{B})$ is a $m times m$ diagonal matrix with positive diagonal elements and $g<0$. Thus, $(gmathbf{B})$ is negative definite.
With $g rightarrow -infty$ I am trying to figure out when the following matrix becomes singular
$$
mathbf{K}^* = mathbf{K} + mathbf{W}(gmathbf{B})mathbf{W}^{T}
$$
The matrix is large and I don't want to calculate e.g. the lowest eigenvalue of the matrix. Instead I'm looking at when the matrix is invertible, thus by the Woodury formula I get
$$
{mathbf{K}^*}^{-1} = mathbf{K}^{-1} - mathbf{K}^{-1} mathbf{W} big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]^{-1} mathbf{W}^T mathbf{K}^{-1}
$$
And I am assuming that the $m times m$ matrix in the brackets equally must be invertible in order for $mathbf{K}^*$ to be invertible. However, I seem to be wrong in this assumption as $mathbf{K}^*$ becomes singular at a different value of $g$ than $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$, so:
- Can I relate the definiteness of $mathbf{K}^*$ to the definiteness of $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$ as $g rightarrow -infty$?
- Can I say anything about when $mathbf{K}^*$ becomes singular based on $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$?
- Any other suggestions on how to determine the point of singularity of $mathbf{K}^*$ other than brute force are welcome.
linear-algebra matrices positive-definite singularity
asked Jan 14 at 12:12
DavidHDavidH
134
$endgroup$
add a comment |
$begingroup$
Consider a real, symmetric and positive definite $ntimes n$ matrix $mathbf{K}$, and a $ntimes m$ matrix $mathbf{W}$. $mathbf{W}$ contains $m$ columns with all zeros except a single entry in each column which is 1. Furthermore $(gmathbf{B})$ is a $m times m$ diagonal matrix with positive diagonal elements and $g<0$. Thus, $(gmathbf{B})$ is negative definite.
With $g rightarrow -infty$ I am trying to figure out when the following matrix becomes singular
$$
mathbf{K}^* = mathbf{K} + mathbf{W}(gmathbf{B})mathbf{W}^{T}
$$
The matrix is large and I don't want to calculate e.g. the lowest eigenvalue of the matrix. Instead I'm looking at when the matrix is invertible, thus by the Woodury formula I get
$$
{mathbf{K}^*}^{-1} = mathbf{K}^{-1} - mathbf{K}^{-1} mathbf{W} big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]^{-1} mathbf{W}^T mathbf{K}^{-1}
$$
And I am assuming that the $m times m$ matrix in the brackets equally must be invertible in order for $mathbf{K}^*$ to be invertible. However, I seem to be wrong in this assumption as $mathbf{K}^*$ becomes singular at a different value of $g$ than $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$, so:
- Can I relate the definiteness of $mathbf{K}^*$ to the definiteness of $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$ as $g rightarrow -infty$?
- Can I say anything about when $mathbf{K}^*$ becomes singular based on $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$?
- Any other suggestions on how to determine the point of singularity of $mathbf{K}^*$ other than brute force are welcome.
linear-algebra matrices positive-definite singularity
asked Jan 14 at 12:12
DavidHDavidH
134
$endgroup$
$begingroup$
"And I am assuming that the m×m matrix in the brackets equally must be invertible in order for K∗ to be invertible." Indeed, you are: addition does horrible things to invertibility: in particular, $K^*$ is invertible when the thing in the brackets is zero. As $gto-infty$, $(gB)^{-1}to0$, so you're really trying to relate the definiteness of $K^*$ and W^TK^{-1}W$. Singularity is probably a stretch: it's a sensitive condition, so a small perturbation would change matters entirely.
$endgroup$
– user3482749
Jan 14 at 12:22
add a comment |
$begingroup$
Consider a real, symmetric and positive definite $ntimes n$ matrix $mathbf{K}$, and a $ntimes m$ matrix $mathbf{W}$. $mathbf{W}$ contains $m$ columns with all zeros except a single entry in each column which is 1. Furthermore $(gmathbf{B})$ is a $m times m$ diagonal matrix with positive diagonal elements and $g<0$. Thus, $(gmathbf{B})$ is negative definite.
With $g rightarrow -infty$ I am trying to figure out when the following matrix becomes singular
$$
mathbf{K}^* = mathbf{K} + mathbf{W}(gmathbf{B})mathbf{W}^{T}
$$
The matrix is large and I don't want to calculate e.g. the lowest eigenvalue of the matrix. Instead I'm looking at when the matrix is invertible, thus by the Woodury formula I get
$$
{mathbf{K}^*}^{-1} = mathbf{K}^{-1} - mathbf{K}^{-1} mathbf{W} big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]^{-1} mathbf{W}^T mathbf{K}^{-1}
$$
And I am assuming that the $m times m$ matrix in the brackets equally must be invertible in order for $mathbf{K}^*$ to be invertible. However, I seem to be wrong in this assumption as $mathbf{K}^*$ becomes singular at a different value of $g$ than $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$, so:
- Can I relate the definiteness of $mathbf{K}^*$ to the definiteness of $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$ as $g rightarrow -infty$?
- Can I say anything about when $mathbf{K}^*$ becomes singular based on $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$?
- Any other suggestions on how to determine the point of singularity of $mathbf{K}^*$ other than brute force are welcome.
linear-algebra matrices positive-definite singularity
asked Jan 14 at 12:12
DavidHDavidH
134
$endgroup$
Consider a real, symmetric and positive definite $ntimes n$ matrix $mathbf{K}$, and a $ntimes m$ matrix $mathbf{W}$. $mathbf{W}$ contains $m$ columns with all zeros except a single entry in each column which is 1. Furthermore $(gmathbf{B})$ is a $m times m$ diagonal matrix with positive diagonal elements and $g<0$. Thus, $(gmathbf{B})$ is negative definite.
With $g rightarrow -infty$ I am trying to figure out when the following matrix becomes singular
$$
mathbf{K}^* = mathbf{K} + mathbf{W}(gmathbf{B})mathbf{W}^{T}
$$
The matrix is large and I don't want to calculate e.g. the lowest eigenvalue of the matrix. Instead I'm looking at when the matrix is invertible, thus by the Woodury formula I get
$$
{mathbf{K}^*}^{-1} = mathbf{K}^{-1} - mathbf{K}^{-1} mathbf{W} big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]^{-1} mathbf{W}^T mathbf{K}^{-1}
$$
And I am assuming that the $m times m$ matrix in the brackets equally must be invertible in order for $mathbf{K}^*$ to be invertible. However, I seem to be wrong in this assumption as $mathbf{K}^*$ becomes singular at a different value of $g$ than $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$, so:
- Can I relate the definiteness of $mathbf{K}^*$ to the definiteness of $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$ as $g rightarrow -infty$?
- Can I say anything about when $mathbf{K}^*$ becomes singular based on $big[ (gmathbf{B})^{-1} + mathbf{W}^T mathbf{K}^{-1} mathbf{W} big]$?
- Any other suggestions on how to determine the point of singularity of $mathbf{K}^*$ other than brute force are welcome.
linear-algebra matrices positive-definite singularity
linear-algebra matrices positive-definite singularity
asked Jan 14 at 12:12
DavidHDavidH
134
asked Jan 14 at 12:12
DavidHDavidH
134
edited Jan 14 at 12:32
DavidH
asked Jan 14 at 12:12
DavidHDavidH
134
asked Jan 14 at 12:12
DavidHDavidH
134
asked Jan 14 at 12:12
DavidHDavidH
134
134
$begingroup$
"And I am assuming that the m×m matrix in the brackets equally must be invertible in order for K∗ to be invertible." Indeed, you are: addition does horrible things to invertibility: in particular, $K^*$ is invertible when the thing in the brackets is zero. As $gto-infty$, $(gB)^{-1}to0$, so you're really trying to relate the definiteness of $K^*$ and W^TK^{-1}W$. Singularity is probably a stretch: it's a sensitive condition, so a small perturbation would change matters entirely.
$endgroup$
– user3482749
Jan 14 at 12:22
add a comment |
$begingroup$
"And I am assuming that the m×m matrix in the brackets equally must be invertible in order for K∗ to be invertible." Indeed, you are: addition does horrible things to invertibility: in particular, $K^*$ is invertible when the thing in the brackets is zero. As $gto-infty$, $(gB)^{-1}to0$, so you're really trying to relate the definiteness of $K^*$ and W^TK^{-1}W$. Singularity is probably a stretch: it's a sensitive condition, so a small perturbation would change matters entirely.
$endgroup$
– user3482749
Jan 14 at 12:22
$begingroup$
"And I am assuming that the m×m matrix in the brackets equally must be invertible in order for K∗ to be invertible." Indeed, you are: addition does horrible things to invertibility: in particular, $K^*$ is invertible when the thing in the brackets is zero. As $gto-infty$, $(gB)^{-1}to0$, so you're really trying to relate the definiteness of $K^*$ and W^TK^{-1}W$. Singularity is probably a stretch: it's a sensitive condition, so a small perturbation would change matters entirely.
$endgroup$
– user3482749
Jan 14 at 12:22
$begingroup$
"And I am assuming that the m×m matrix in the brackets equally must be invertible in order for K∗ to be invertible." Indeed, you are: addition does horrible things to invertibility: in particular, $K^*$ is invertible when the thing in the brackets is zero. As $gto-infty$, $(gB)^{-1}to0$, so you're really trying to relate the definiteness of $K^*$ and W^TK^{-1}W$. Singularity is probably a stretch: it's a sensitive condition, so a small perturbation would change matters entirely.
$endgroup$
– user3482749
Jan 14 at 12:22
add a comment |
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$begingroup$
"And I am assuming that the m×m matrix in the brackets equally must be invertible in order for K∗ to be invertible." Indeed, you are: addition does horrible things to invertibility: in particular, $K^*$ is invertible when the thing in the brackets is zero. As $gto-infty$, $(gB)^{-1}to0$, so you're really trying to relate the definiteness of $K^*$ and W^TK^{-1}W$. Singularity is probably a stretch: it's a sensitive condition, so a small perturbation would change matters entirely.
$endgroup$
– user3482749
Jan 14 at 12:22