I would like an equation of a matrix pseudo-inverse to be explained
I am currently reading a paper titled "A Noise Tolerant Algorithm for Wrist-Mounted Robotic Sensor Calibration with or without Sensor Orientation Measurement", i can email you a snapshot of this paper since i bought it off IEEE.
In this paper, an equation of the form $AX = B$ is solved for $X$ using what appears to be the pseudo-inverse of $A$ (denoted as $A^+$), and then it is stated that:
$$A^+ = (A^TA)^{-1}(A^T)$$
I am not able to follow how this equality is true, is it?
Thank you,
linear-algebra matrices matrix-equations pseudoinverse
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
asked Dec 8 at 16:35
Jad Tawil
11
add a comment |
I am currently reading a paper titled "A Noise Tolerant Algorithm for Wrist-Mounted Robotic Sensor Calibration with or without Sensor Orientation Measurement", i can email you a snapshot of this paper since i bought it off IEEE.
In this paper, an equation of the form $AX = B$ is solved for $X$ using what appears to be the pseudo-inverse of $A$ (denoted as $A^+$), and then it is stated that:
$$A^+ = (A^TA)^{-1}(A^T)$$
I am not able to follow how this equality is true, is it?
Thank you,
linear-algebra matrices matrix-equations pseudoinverse
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
asked Dec 8 at 16:35
Jad Tawil
11
3
What definition of the pseudo-inverse are you working with? Are you asking us to show you how this formula for $A^{+}$ satisfies that definition?
– Brian Borchers
Dec 8 at 17:22
Assuming that $A$ has full column rank, left-multiply both sides of the matrix equation by $A^top$, then left-multiply both sides by the inverse of $A^top A$.
– Rodrigo de Azevedo
Dec 8 at 17:24
According to Wikipedia, there are four defining properties of pseudo-inverse and it always exists and is unique. When $A^intercal A$ is invertible, we can show that $(A^intercal A)^{-1} A^intercal$ satisfies the four defining properties mentioned above. Therefore, we can write $A^+ = (A^intercal A)^{-1}A^intercal$ (since it exists and is unique as mentioned above).
– Alex Vong
Dec 8 at 17:52
@BrianBorchers apologies for the ambiguity, it is assumed to be a Moore-Penrose inverse.
– Jad Tawil
Dec 9 at 19:55
add a comment |
I am currently reading a paper titled "A Noise Tolerant Algorithm for Wrist-Mounted Robotic Sensor Calibration with or without Sensor Orientation Measurement", i can email you a snapshot of this paper since i bought it off IEEE.
In this paper, an equation of the form $AX = B$ is solved for $X$ using what appears to be the pseudo-inverse of $A$ (denoted as $A^+$), and then it is stated that:
$$A^+ = (A^TA)^{-1}(A^T)$$
I am not able to follow how this equality is true, is it?
Thank you,
linear-algebra matrices matrix-equations pseudoinverse
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
asked Dec 8 at 16:35
Jad Tawil
11
I am currently reading a paper titled "A Noise Tolerant Algorithm for Wrist-Mounted Robotic Sensor Calibration with or without Sensor Orientation Measurement", i can email you a snapshot of this paper since i bought it off IEEE.
In this paper, an equation of the form $AX = B$ is solved for $X$ using what appears to be the pseudo-inverse of $A$ (denoted as $A^+$), and then it is stated that:
$$A^+ = (A^TA)^{-1}(A^T)$$
I am not able to follow how this equality is true, is it?
Thank you,
linear-algebra matrices matrix-equations pseudoinverse
linear-algebra matrices matrix-equations pseudoinverse
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
asked Dec 8 at 16:35
Jad Tawil
11
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
asked Dec 8 at 16:35
Jad Tawil
11
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
edited Dec 8 at 17:25
Rodrigo de Azevedo
12.8k41854
12.8k41854
asked Dec 8 at 16:35
Jad Tawil
11
asked Dec 8 at 16:35
Jad Tawil
11
asked Dec 8 at 16:35
Jad Tawil
11
11
3
What definition of the pseudo-inverse are you working with? Are you asking us to show you how this formula for $A^{+}$ satisfies that definition?
– Brian Borchers
Dec 8 at 17:22
Assuming that $A$ has full column rank, left-multiply both sides of the matrix equation by $A^top$, then left-multiply both sides by the inverse of $A^top A$.
– Rodrigo de Azevedo
Dec 8 at 17:24
According to Wikipedia, there are four defining properties of pseudo-inverse and it always exists and is unique. When $A^intercal A$ is invertible, we can show that $(A^intercal A)^{-1} A^intercal$ satisfies the four defining properties mentioned above. Therefore, we can write $A^+ = (A^intercal A)^{-1}A^intercal$ (since it exists and is unique as mentioned above).
– Alex Vong
Dec 8 at 17:52
@BrianBorchers apologies for the ambiguity, it is assumed to be a Moore-Penrose inverse.
– Jad Tawil
Dec 9 at 19:55
add a comment |
3
What definition of the pseudo-inverse are you working with? Are you asking us to show you how this formula for $A^{+}$ satisfies that definition?
– Brian Borchers
Dec 8 at 17:22
Assuming that $A$ has full column rank, left-multiply both sides of the matrix equation by $A^top$, then left-multiply both sides by the inverse of $A^top A$.
– Rodrigo de Azevedo
Dec 8 at 17:24
According to Wikipedia, there are four defining properties of pseudo-inverse and it always exists and is unique. When $A^intercal A$ is invertible, we can show that $(A^intercal A)^{-1} A^intercal$ satisfies the four defining properties mentioned above. Therefore, we can write $A^+ = (A^intercal A)^{-1}A^intercal$ (since it exists and is unique as mentioned above).
– Alex Vong
Dec 8 at 17:52
@BrianBorchers apologies for the ambiguity, it is assumed to be a Moore-Penrose inverse.
– Jad Tawil
Dec 9 at 19:55
3
3
What definition of the pseudo-inverse are you working with? Are you asking us to show you how this formula for $A^{+}$ satisfies that definition?
– Brian Borchers
Dec 8 at 17:22
What definition of the pseudo-inverse are you working with? Are you asking us to show you how this formula for $A^{+}$ satisfies that definition?
– Brian Borchers
Dec 8 at 17:22
Assuming that $A$ has full column rank, left-multiply both sides of the matrix equation by $A^top$, then left-multiply both sides by the inverse of $A^top A$.
– Rodrigo de Azevedo
Dec 8 at 17:24
Assuming that $A$ has full column rank, left-multiply both sides of the matrix equation by $A^top$, then left-multiply both sides by the inverse of $A^top A$.
– Rodrigo de Azevedo
Dec 8 at 17:24
According to Wikipedia, there are four defining properties of pseudo-inverse and it always exists and is unique. When $A^intercal A$ is invertible, we can show that $(A^intercal A)^{-1} A^intercal$ satisfies the four defining properties mentioned above. Therefore, we can write $A^+ = (A^intercal A)^{-1}A^intercal$ (since it exists and is unique as mentioned above).
– Alex Vong
Dec 8 at 17:52
According to Wikipedia, there are four defining properties of pseudo-inverse and it always exists and is unique. When $A^intercal A$ is invertible, we can show that $(A^intercal A)^{-1} A^intercal$ satisfies the four defining properties mentioned above. Therefore, we can write $A^+ = (A^intercal A)^{-1}A^intercal$ (since it exists and is unique as mentioned above).
– Alex Vong
Dec 8 at 17:52
@BrianBorchers apologies for the ambiguity, it is assumed to be a Moore-Penrose inverse.
– Jad Tawil
Dec 9 at 19:55
@BrianBorchers apologies for the ambiguity, it is assumed to be a Moore-Penrose inverse.
– Jad Tawil
Dec 9 at 19:55
add a comment |
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3
What definition of the pseudo-inverse are you working with? Are you asking us to show you how this formula for $A^{+}$ satisfies that definition?
– Brian Borchers
Dec 8 at 17:22
Assuming that $A$ has full column rank, left-multiply both sides of the matrix equation by $A^top$, then left-multiply both sides by the inverse of $A^top A$.
– Rodrigo de Azevedo
Dec 8 at 17:24
According to Wikipedia, there are four defining properties of pseudo-inverse and it always exists and is unique. When $A^intercal A$ is invertible, we can show that $(A^intercal A)^{-1} A^intercal$ satisfies the four defining properties mentioned above. Therefore, we can write $A^+ = (A^intercal A)^{-1}A^intercal$ (since it exists and is unique as mentioned above).
– Alex Vong
Dec 8 at 17:52
@BrianBorchers apologies for the ambiguity, it is assumed to be a Moore-Penrose inverse.
– Jad Tawil
Dec 9 at 19:55