This lecture specifies the concept of heat equivalence and also proves part propositions about row indistinguishable matrices the lie at the love of numerous important outcomes in direct algebra.

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Definition let and be two matrices. We say the is row identical to if and also only if over there exist primary school matrices " style="background-position:0px -686px;vertical-align:-5px"/> such that" />


Remember that pre-multiplying by one elementary matrix is the very same as performing one primary school row operation on . Therefore, is row equivalent to if and only if can be transformed right into by performing a sequence of elementary row operations on .

Equivalence relation

row equivalence is an equivalence relation because it is:

symmetric: if is row identical to , then is row indistinguishable to ;

transitive: if is equivalent to and also is identical to , climate is tantamount to ;

reflexive: is equivalent to itself.


Proof

Suppose is row indistinguishable to . Since an elementary matrix is invertible and also its inverse is an elementary matrix, we have actually that

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" />where " style="background-position:0px -554px;vertical-align:-5px"/> space elementary matrices. Therefore, is indistinguishable to . This proves symmetry. If is indistinguishable to and is tantamount to , then" />and
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" />where " style="background-position:0px -686px;vertical-align:-5px"/> and " style="background-position:0px -711px;vertical-align:-5px"/> room elementary matrices. Now, pre-multiply both sides of the first equation by " style="background-position:0px -662px;vertical-align:-5px"/>:
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" />Then, is equivalent to , that is, row equivalence is transitive. Finally, for any type of elementary procession , we can write" style="background-position:0px -103px;"/>Since is elementary, this method that we can transform into itself by means of elementary heat operations. As a consequence, heat equivalence is reflexive.


Column post property

The next proposition states crucial property of row equivalence, known as tower correspondence property.


Proposition let and also be two matrices. Permit it is in row indistinguishable to . Signify by and also the -th columns the and also respectively. Then, " style="background-position:0px -580px;"/>for an vector if and only if" style="background-position:0px -621px;"/>


Proof

Since is row tantamount to , we have actually that" style="background-position:0px -735px;"/>where is a product of elementary school matrices:" style="background-position:0px -361px;"/>Furthermore, by the very meaning of procession product (see also here):" style="background-position:0px -478px;"/>Thus, we have the right to substitute" style="background-position:0px -516px;"/>and" style="background-position:0px -440px;"/>in the equation" style="background-position:0px -580px;"/>so regarding obtain" style="background-position:0px -1px;"/>By pre-multiplying both political parties by , us get" style="background-position:0px -621px;"/>Thus, we have actually proved the implies . The opposite implication ( suggests ), can be proved analogously.


In other words, as soon as and are row equivalent, the -th obelisk of can be written as a linear mix of a given collection of columns that itself, with coefficients taken indigenous the vector , if and also only if the -th column of is a linear combination of the corresponding collection of columns the , through coefficients taken from the very same vector .

A advantageous corollary of the previous proposition follows.


Proposition let and also be 2 row equivalent matrices. Then, a collection of columns the is linearly independent if and also only if the corresponding set of columns that is linearly independent.


Proof

The evidence is through contradiction. Expect that a set of columns the is linearly independent, however the matching columns the room linearly dependent. It adheres to that a column deserve to be composed as a linear combination of various other columns:" style="background-position:0px -621px;"/>where . In particular, there space some non-zero entries the corresponding to the columns in the collection we room considering. But by the vault proposition, this suggests that " style="background-position:0px -580px;"/>In various other words, the collection of columns of is not linearly independent, a contradiction. Therefore, if a collection of columns that is linearly independent, then the corresponding columns of need to be linearly independent. The contrary implication deserve to be showed analogously.


Dominant columns

This section introduces the principle of leading columns, which will certainly be used below to research the nature of row equivalent matrices.


Definition let be a matrix. Represent its -th column by . We say that is a dominant column if and only if it cannot be composed as a linear mix of the columns come its left.


A very first simple an outcome about dominant columns follows.


Proposition Two indistinguishable matrices and also have the same set of dominant columns, that is, the set of indexes of the dominant columns that corresponds with the collection of indexes of the dominant columns that .


Proof

Suppose is a leading column the . Then, over there is no vector

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" />such the " style="background-position:0px -580px;"/>By the shaft correspondence building above, this is possible if and also only if there is no such vector satisfying" style="background-position:0px -621px;"/>As a consequence, cannot be composed as a linear mix of the columns to its left. Therefore it is dominant. We have just proved that is dominant only if is dominant. The evidence of the opposite implication is analogous. This hold for any . Therefore, the columns the are leading in room dominant likewise in and also vice versa.


because that instance, if the dominant columns of are the second, 3rd and fifth, climate the leading columns of are the second, third and fifth.

Row tantamount matrices in diminished row echelon form

The propositions above allow us to prove part properties of matrices in decreased row echelon form.

Remember the a procession is in reduced row echelon type (RREF) if and also only if:

all its non-zero rows save an element, dubbed pivot, the is equal to 1 and has only zero entries in the quadrant below it and also to its left;

every pivot is the only non-zero aspect in its column;

all the zero rows (if there are any) are listed below the non-zero rows.

Furthermore, the Gauss-Jordan removed algorithm have the right to be supplied to transform any type of matrix right into an RREF procession by elementary row operations. Therefore, any kind of matrix is row equivalent to one RREF matrix.

Remember that a an easy column is a obelisk containing a pivot, while a non-basic pillar does no contain any kind of pivot.

The simple columns of one RREF matrix room vectors the the canonical basis, that is, they have actually one entry same to 1 and also all the various other entries same to zero. Furthermore, if an RREF matrix has straightforward columns, climate those columns are the an initial vectors the the canonical basis, as declared by the following proposition.


Proposition let it is in a procession in diminished row echelon form. Then, the -th an easy column the , counting from the left, is same to the -th vector the the canonical basis, that is, it has actually a 1 in position and also all its various other entries space equal come 0.


Proof

By the definition of RREF matrix, the an easy columns of space vectors of the canonical communication (they have one entry equal to 1 and also all various other entries same to 0). Furthermore, every non-zero rows save on computer a pivot. Therefore, the -th simple column has the -th pivot, i beg your pardon is situated on the -th row. In various other words, the pivot, i m sorry is equal to 1, is the -th entrance of the -th straightforward column.


If a tower is non-basic, that is, it has no pivot, then it can be written as" style="background-position:0px -179px;"/>where is the variety of basic columns come its left (the entries below the -th must be zero due to the fact that the -th pivot, with k$" style="background-position:0px -955px;vertical-align:-5px"/>, has only 0s come its left). Therefore, the non-basic column can be written as a linear mix of the columns to its left. For example, if and the first, 3rd and 4th columns are basic, then

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" />Thus, if a column is non-basic the is not linearly independent from the columns come its left. Hence, it is no a dominant column.


By combine the two an easy propositions above, we gain the following one.


Proposition If a matrix is in reduced row echelon form, then one of its columns is an easy if and only if that is dominant, and also it is non-basic if and only if it is not dominant.


Proof

We have currently explained that any kind of matrix is row tantamount to a matrix in diminished row echelon type which can be obtained by making use of the Gauss-Jordan elimination algorithm. We should prove uniqueness. Expect that 2 matrices and room in lessened row echelon kind and the they room both row tantamount to . Due to the fact that row equivalence is transitive and also symmetric, and also space row equivalent. Therefore, the positions of their leading columns coincide. Equivalently, the location of their simple columns coincide. Yet we have actually proved over that the -th simple column of an RREF matrix, counting indigenous the left, is equal to the -th vector the the canonical basis. Therefore, not only the basic columns of and also have the very same positions, but their corresponding entries coincide. The non-basic columns are direct combinations that the straightforward ones. By the tower correspondence building above, the coefficients of the straight combinations are the exact same for and . But likewise the vectors being merged linearly coincide since the simple columns that and coincide. As a consequence, every non-basic column of is equal to the corresponding non-basic shaft of . Thus, , i beg your pardon proves that the row tantamount RREF the a procession is unique.


A an effect of this uniqueness result is the if two matrices space row equivalent, climate they are equivalent to the very same RREF matrix.


Proposition permit be row identical to . Then, and are identical to the exact same RREF matrix .


Proof

Denote by and the RREF matrices that space row indistinguishable to and also respectively:" style="background-position:0px -39px;"/>where and also are assets of elementary school matrices. Furthermore, is row tantamount to , therefore that" style="background-position:0px -141px;"/>where is a product of elementary matrices. Us pre-multiply both sides of eq. (3) by , so as to get

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" />Since is a product of elementary school matrices, is an RREF procession row tantamount to . But the RREF row tantamount matrix is unique. Therefore, .


Proposition let be a invertible matrix. Then, is row identical to the identity matrix .


By the results above, we know that is row equivalent to a distinctive RREF procession . Furthermore, can be transformed right into by elementary row operations, the is, through pre-multiplying by an invertible procession (equal to the product the the primary school matrices supplied to do the row operations):" style="background-position:0px -773px;"/>But we understand that pre-multiplication by an invertible (i.e., full-rank) procession go not change the rank. Therefore, is full-rank. Together a consequence, all the columns that are simple (there cannot be non-basic columns because the columns the a full-rank matrix room all linearly elevation from every other). Yet this method that the columns that space the vectors the the canonical basis of the room of -dimensional vectors. In various other words, they room the columns that the identity matrix. Hence, .


Clearly, due to the fact that the identity matrix is a matrix in diminished row echelon form, any kind of invertible matrix is tantamount to the distinct RREF matrix .

one immediate repercussion of the previous proposition follows.


Proposition allow be a invertible matrix. Then, deserve to be created as a product of primary school matrices:" style="background-position:0px -399px;"/>where " style="background-position:0px -686px;vertical-align:-5px"/> space elementary matrices.


Proof

By the previous proposition, the identification matrix is row tantamount to . So, through the definition of row identical matrix, we have that

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" />where " style="background-position:0px -686px;vertical-align:-5px"/> are elementary matrices.


Proposition allow it is in an RREF procession that is row tantamount to a procession . Climate and also have the same rank. The rank is equal to 1) the variety of non-zero rows the or, equivalently, come 2) the variety of basic columns the .


First that all, remember that pre-multiplying a matrix by an invertible procession does not readjust the location of . As a consequence, if (an invertible product of elementary school matrices) transforms right into a row equivalent RREF matrix , we have actually that

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" />The location of is equal to the maximum number of linearly elevation columns that . The simple columns of room linearly independent, while the non-basic columns have the right to be written as direct combinations that the basic ones. Therefore, the rank of is same to the number of basic columns of . Furthermore, each an easy columns has a pivot and each non-zero row includes a pivot. Together a consequence, the location is likewise equal come the number of non-zero rows of .

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How to cite

Please mention as:

Taboga, Marco (2017). "Row equivalence", Lectures on procession algebra. Https://www.urbanbreathnyc.com/matrix-algebra/row-equivalence.