Witness (mathematics)

Given an $m\times n$ matrix $A$ of rank $r$, a rank decomposition or rank factorization of $A$ is a product $A=CF$, where $C$ is an $m\times r$ matrix and $F$ is an $r\times n$ matrix.

Every finite dimensional matrix has a rank decomposition: Let $A$ be an $m\times n$ matrix whose column rank is $r$. Therefore, there are $r$ linearly independent columns in $A$; equivalently, the dimension of the column space of $A$ is $r$. Let $c_1,c_2,\dots ,c_r$ be any basis for the column space of $A$ and place them as column vectors to form the $m\times r$ matrix $C=[c_1:c_2:\dots :c_r]$. Therefore, every column vector of $A$ is a linear combination of the columns of $C$. To be precise, if $A=[a_1:a_2:\dots :a_n]$ is an $m\times n$ matrix with $a_j$ as the $j$-th column, then

$a_j=f_{1j}{c}_1+f_{2j}{c}_2+\cdots +f_{rj}{c}_r,$

where $f_{ij}$'s are the scalar coefficients of $a_j$ in terms of the basis $c_1,c_2,\dots ,c_r$. This implies that $A=CF$, where $f_{ij}$ is the $(i,j)$-th element of $F$.

rank($A$) = rank($A^{\text{T}}$)

An immediate consequence of rank factorization is that the rank of $A$ is equal to the rank of its transpose $A^{\text{T}}$. Since the columns of $A$ are the rows of $A^{\text{T}}$, the column rank of $A$ equals its row rank.

Proof: To see why this is true, let us first define rank to mean column rank. Since $A=CF$, it follows that $A^{\text{T}}=F^{\text{T}}{C}^{\text{T}}$. From the definition of matrix multiplication, this means that each column of $A^{\text{T}}$ is a linear combination of the columns of $F^{\text{T}}$. Therefore, the column space of $A^{\text{T}}$ is contained within the column space of $F^{\text{T}}$ and, hence, rank($A^{\text{T}}$) ≤ rank($F^{\text{T}}$). Now, $F^{\text{T}}$ is $n$×$r$, so there are $r$ columns in $F^{\text{T}}$ and, hence, rank($A^{\text{T}}$) ≤ $r$ = rank($A$). This proves that rank($A^{\text{T}})$ ≤ rank($A$). Now apply the result to $A^{\text{T}}$ to obtain the reverse inequality: since $(A^{\text{T}}{)}^{\text{T}}$ = $A$, we can write rank($A$) = rank($(A^{\text{T}}{)}^{\text{T}})$ ≤ rank($A^{\text{T}}$). This proves rank($A)$ ≤ rank($A^{\text{T}}$). We have, therefore, proved rank($A^{\text{T}})$ ≤ rank($A$) and rank($A$) ≤ rank($A^{\text{T}}$), so rank($A$) = rank($A^{\text{T}}$). (Also see the first proof of column rank = row rank under rank).

Rank Factorization from Row Echelon Forms

In practice, we can construct one specific rank factorization as follows: we can compute $B$, the reduced row echelon form of $A$. Then $C$ is obtained by removing from $A$ all non-pivot columns, and $F$ by eliminating all zero rows of $B$.

Example

Consider the matrix

$A=\left[\begin{array}{cccc}1& 3& 1& 4\\ 2& 7& 3& 9\\ 1& 5& 3& 1\\ 1& 2& 0& 8\end{array}\right]\sim \left[\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\end{array}\right]=B\text{.}$

$B$ is in reduced echelon form. Then $C$ is obtained by removing the third column of $A$, the only one which is not a pivot column, and $F$ by getting rid of the last row of zeroes, so

$C=\left[\begin{array}{ccc}1& 3& 4\\ 2& 7& 9\\ 1& 5& 1\\ 1& 2& 8\end{array}\right]\text{,}F=\left[\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]\text{.}$

It is straightforward to check that

$A=\left[\begin{array}{cccc}1& 3& 1& 4\\ 2& 7& 3& 9\\ 1& 5& 3& 1\\ 1& 2& 0& 8\end{array}\right]=\left[\begin{array}{ccc}1& 3& 4\\ 2& 7& 9\\ 1& 5& 1\\ 1& 2& 8\end{array}\right]\left[\begin{array}{cccc}1& 0& -2& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right]=CF\text{.}$

Proof

Let $P$ be an $n\times n$ permutation matrix such that $AP=(C,D)$ in block partitioned form, where the columns of $C$ are the $r$ pivot columns of $A$. Every column of $D$ is a linear combination of the columns of $C$, so there is a matrix $G$ such that $D=CG$, where the columns of $G$ contain the coefficients of each of those linear combinations. So $AP=(C,CG)=C(I_r,G)$, $I_r$ being the $r\times r$ identity matrix. We will show now that $(I_r,G)=FP$.

Transforming $AP$ into its reduced row echelon form amounts to left-multiplying by a matrix $E$ which is a product of elementary matrices, so $EAP=BP=EC(I_r,G)$, where $EC=\left(\begin{array}{c}{I}_r\\ 0\end{array}\right)$. We then can write $BP=\left(\begin{array}{cc}{I}_r& G\\ 0& 0\end{array}\right)$, which allows us to identify $(I_r,G)=FP$, i.e. the nonzero $r$ rows of the reduced echelon form, with the same permutation on the columns as we did for $A$. We thus have $AP=CFP$, and since $P$ is invertible this implies $A=CF$, and the proof is complete.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010