# Matrix and there is a nonzero a such that xa0 then x

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matrix and there is a nonzero a such that Xa=0 then X is column-singular . The columns of X are said to be linearly dependent . If Xa=0 if and only if a=0 , then X is column-non- singular . The columns are linearly independent . Row-singularity and row-non-singularity are defined in the same way. X is row-singular if and only if X' is column-singular . 48

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If X is column-singular then (X|Y) is column- singular for all Y . If X is column-non-singular, then (X|Y) is column-singular if and only if there is a B such that Y=XB . If X is broad, then X is column-singular. A square matrix is non-singular if it is row-non- singular and column-non-singular. A square matrix is singular if it is not non-singular. 49
Suppose A is triangular, with non-zero diagonal. Then A is non-singular. If then it follows that x 1 =0 , and thus x 2 =0 , and thus x 3 =0 . 50 a 11 0 0 a 21 a 22 0 a 31 a 32 a 33 x 1 x 2 x 3 = 0 0 0

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Suppose A is non-singular. Then YA is column- non-singular if an only if Y is column-non- singular. Proof: If Y is column-non-singular then YAb=0 if and only if Ab=0 if and only if b=0 . 51
The column-rank of an n x m matrix X is equal to p if there is an n x p matrix column-non-singular Y and an m x p column-non-singular matrix A such that X=YA' . Such a decomposition is called a full-rank decomposition. Thus the m columns of X are linearly independent combinations of p linearly independent columns. 52

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Here our discussion of the full-rank decomposition is not constructive, we merely assert its existence and use it to define rank. Later on we shall discuss the QR-decomposition and the s ingular value decomposition , which are both constructive examples of full rank decompositions . 53
The column-nullity of an n x m matrix X is m minus the column-rank of X . Row-rank and row-nullity are defined in a similar way (and course the row-rank of X is the column- rank of X' ). Clearly the row-rank and the column-rank of X are equal: if X=YA' then X'=AY' . Thus we can unambiguously speak of the rank of a matrix X. A square matrix is of full rank if it is non-singular . 54

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A column-singular n x m matrix X has rank less than m and column-nullity larger than zero. The n x m matrix X is of full column-rank if its rank is m , i.e. if its columns are linearly independent. If the n x m tall matrix X is of full column-rank , then there is an n x (n -m) tall matrix Y of full column rank such that (X|Y) is non-singular. If X=0 then the rank of X is 0 . 55
Pre-multiplying or post-multiplying a matrix X by a square non-singular matrix does not change it's rank or nullity. Thus if S and T are non- singular we have rank(SXT)=rank(X) . Proof: if X=AY' is a full-rank decomposition, then so is SXT=(SA)(Y'T). The matrices X , X'X , and XX' have the same rank. Proof: If X=YA' is a full rank decomposition then so are X'X=A(Y'YA') and XX'=Y(A'AY') .

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