rank of product of matriceseigenvalues of adjacency matrix
Written by on November 16, 2022
column vector
The effect of changing one of the matrix of matrix multiplication on the rank of the resulting new matrix? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Start a research project with a student in my class. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Proposition
That is, we have Similarly, we have and since . Justify your answer. Most of the learning materials found on this website are now available in a traditional textbook format. Given two matrices $A_{m\times n}$ and $B_{n\times p}$, what is the sufficient and necessary condition for $AB$ to have full rank?
How can I output different data from each line? Somewhat expanding the comments, one can say the following. Stack Overflow for Teams is moving to its own domain! matrix and
a square
If $\text{Rank}(\mathbf{A}) = \text{Rank}(\mathbf{B}) = m$ then what is the rank of $\mathbf{ABA}^\top$? To learn more, see our tips on writing great answers. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. GCC to make Amiga executables, including Fortran support? I'm interested in the rank of the matrix How difficult would it be to reverse engineer a device whose function is based on unknown physics? that
Now what i don't know is weather this is true in general, as what i saw in general is the inequality, but don't know if the equality hold to. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. . the space generated by the columns of
Shrinkwrap modifier leaving small gaps when applied. rev2022.11.15.43034. is full-rank. Apparently this is a corollary to the theorem If A and B are two matrices which can be multiplied, then rank (AB) <= min ( rank (A), rank (B) ).
Is there a penalty to leaving the hood up for the Cloak of Elvenkind magic item? is a linear combination of the rows of
Extremely clear example!
Here is my question: Given $A_{m \times n}$ matrix with rank $m$, and $B_{n \times p}$ matrix with rank $p$, where $n > p \geq m$. $$
Sci-fi youth novel with a young female protagonist who is watching over the development of another planet. then. ,
,
Asking for help, clarification, or responding to other answers. Assume that $A\in\mathbb{R}^{n\times \nu}$, $B\in\mathbb{R}^{\nu\times \nu}$ and $C\in\mathbb{R}^{\nu\times n}$.
Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Please your help on this is highly welcomed. This implies z R r such that. Making statements based on opinion; back them up with references or personal experience. Let
I have to find the actual rank of the product matrix AB. . To learn more, see our tips on writing great answers. thenso
Thus, we have proved that the space spanned by the columns of
Proving that $p = \inf\{\|Ax-b\|: x\in\mathbb{R}^n\}$ is attained, Proving that $\operatorname{rank}(AB)$ is smaller or equal to $\operatorname{rank}(B)$, Proving that there is no matrix $B$ for such $By = x$. Proposition
Thus, the only vector that
Prove that if
is there any mistake currently in this answer? Now what i don't know is weather this is true in general, as what i saw in general is the inequality, but don't know if the equality hold to. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. and
is an
is called a Gram matrix. "Matrix product and rank", Lectures on matrix algebra. linear-algebra matrices matrix-rank 13,883 Solution 1 Assuming that N (the number of vectors) is the same as n (the dimension of the space) then yes. When $n>\max(m,p)$ all one can say in general, knowing just the ranks of $A$ and $B$, is the usual necessary condition either that $B$ be injective (if $m\geq p$) or that $A$ be surjective (if $m\leq p$) in order for $AB$ to be injective respectively surjective, that is, full rank. Solved Examples on Rank of Matrix Example 1: Find the rank of the matrix [ 1 2 3 2 3 4 3 5 7] . . Why do paratroopers not get sucked out of their aircraft when the bay door opens? When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. As a consequence, there exists a
Connect and share knowledge within a single location that is structured and easy to search. is the
Actual/Full Matrix B is given which is of size 3 * 3 and I am able to find that the rank of B is also 3. Then $A^TBC = 0$ but $A^T B^{-1}C = 1.$ Here the matrix $B$ was carefully crafted to rotate the image of $C$ into the kernel of $A^T$, but not vice-versa. is the space
@Srivatsan: I don't see how the answer can be directly translated to my question, unless you show explicitly. rev2022.11.15.43034. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Does $ \mbox{rank} (A B) \leq \min \left( \mbox{rank} (A), \mbox{rank}(B) \right)$ hold with probability $1$ when matrices $A$ and $B$ are random? Then its determinant is calculated as the product of the principal . Moreover, the rows of
Is atmospheric nitrogen chemically necessary for life? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. Any vector
If $A \in \mathbb{C}^{m\times n}$ is full-column rank matrix, then is rank($AB$) = rank ($BA$) = rank($B$)? \operatorname{rank}(AB) =
$A$ and $C$ are full rank ($n$), and of course, $B$ is as well ($\nu$). It is left as an exercise (see
Portable Object-Oriented WC (Linux Utility word Count) C++ 20, Counts Lines, Words Bytes.
How to handle. Making statements based on opinion; back them up with references or personal experience. can be written as a linear combination of the columns of
is the rank of
Consider nonzero v 1 span { u 2, , u N } .
we
So let's add these two matrices together first. Then, Proof I did create Matrix A and B and find the rank of the Product, it gives me the smaller of rank(A) and rank(B). To see this, note that for any vector of coefficients
,
Possible Duplicate: In addition to multiplying a matrix by a scalar, we can multiply two matrices.
be a
Being full-rank, both matrices have rank
The best answers are voted up and rise to the top, Not the answer you're looking for? means that any
Thanks. Thanks again!
How did knights who required glasses to see survive on the battlefield? $$\mathrm{rank} AB=\mathrm{rank} A'B=p=\min\{n,p\}.$$. (b) If the matrix B is nonsingular, then rank ( A B) = rank ( A). vector (being a product of an
a square
thenso
can be written as a linear combination of the columns of
Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. MathJax reference.
two matrices are equal. be a
such
You want to prove that if A is an M by n matrix and B is an n by n matrix of rank n, then rank (AB) = rank (A). Thanks for contributing an answer to Mathematics Stack Exchange! are
for any vector of coefficients
Use MathJax to format equations. prove that the rank of matrix $AB$ is at most $\mathrm{rank}(A)$. Do solar panels act as an electrical load on the sun? What does 'levee' mean in the Three Musketeers? such
which means rank (AB) = min (rank (A),rank (B)).
vector of coefficients of the linear combination. What have you tried yourself to prove it?
Let where and are column vectors of and , respectively. So we have one plus five two plus six, three plus seven and four plus eight. is a linear combination of the rows of
"Cropping" the resulting shared secret from ECDH. is the rank of
rank (ab)min (rank (a),rank (b)) properties of rank of product of matrices and its transpose 3,351 views Mar 1, 2021 51 Dislike Share Save linear algebra 4.51K subscribers 1. Connect and share knowledge within a single location that is structured and easy to search. Why did The Bahamas vote against the UN resolution for Ukraine reparations? rank of the
The best answers are voted up and rise to the top, Not the answer you're looking for? The term rank of A will be denoted by (A). In light of this operation I'm wondering if: Let me answer you second question first: unfortunately no, the conditions are not equivalent.
matrix and an
are full-rank. haveThe
and
Connect and share knowledge within a single location that is structured and easy to search. Proposition
whose dimension is
$$AB^{-1}C.$$
. Now consider the matrix A T A. Then, the product
that can be written as linear
Taboga, Marco (2021). and that spanned by the columns of
Calculate difference between dates in hours with closest conditioned rows per group in R. Why do many officials in Russia and Ukraine often prefer to speak of "the Russian Federation" rather than more simply "Russia"? We can also
$B$ is (nxm) matrix with rank k. I want to prove analytically that $$rk (AB) = k$$ I am not sure if my answer is correct or not, but here it is: ,
Conclusion: True! In light of this operation I'm wondering if: is the
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
Let
( 4 2 8 4 4 2 ) . multiply it by a full-rank matrix.
We assume without loss of generality that all matrices considered have no rows or columns consisting entirely of 0's. Type Research Article Information Canadian Journal of Mathematics , Volume 18 , 1966 , pp.
Muthuraja M Asks: Finding Rank of Product of Matrices Size and Rank of Matrix A is given as 10 * 3 and 3 respectively. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. the exercise below with its solution). if we take A and B be 2 non-zero matrices s.t AB= zero matrix then rank A,rank B >0 but rank of AB = 0 so rank AB is not equal to min{rank A, rank B}. $$ Toilet supply line cannot be screwed to toilet when installing water gun. matrix. coincide. then. What clamp to use to transition from 1950s-era fabric-jacket NM? haveNow,
. ,
which implies that the columns of
Rank (AB) Rank (A) + Rank (B) Rank (AB) n 2. ,
coincide. is full-rank, it has less columns than rows and, hence, its columns are
entry of the
and that spanned by the rows of
Asking for help, clarification, or responding to other answers. One of the important theorems one learns in linear algebra is that N u l ( A T) = C o l ( A), N u l ( A) = C o l ( A T). This was very helpful. when does $\operatorname{rank}(AB)=\operatorname{rank}(A)$?
The rank of $AB$ is equal to the dimension of the image of $AB,$ and similarly for the rank of $A.$ The image of $A$ contains the image of $AB.$, $$\mathcal{C}(A) = \{Ax | x \in \mathbb{R}^n\}$$, $$\mathcal{C}(AB) = \{ABy | y \in \mathbb{R}^r\}$$, Assume to the contrary that the rank of the latter is strictly greater than the first. We claim that Any vector can be written as shani Asks: Rank of product of matrices: $rank(ABA') = rank(A)$ when rank(A) = rank(B) Suppose $\mathbf{A}$ is an $m \times n$ matrix, $\mathbf{B}$ is a symmetric $n \times n$ matrix, and $m < n$. Assume we want to multiply two matrices $A$ and $B$, and we want to calculate the rank. The above is closer to the actual question generalised.
thatThusWe
Have you tried to find matrices $A$ and $B$ that serve as a counterexample? If
is full-rank,
Handled. Square matrix A square matrix is a . -th
vector of coefficients of the linear combination.
Doctor of Mathematics. How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? givesis
As a consequence, also their dimensions coincide. denotes the
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What do we mean when we say that black holes aren't made of anything? If all strict submatrices have full rank, does the matrix have full rank? What is an idiom about a stubborn person/opinion that uses the word "die"? matrix.
. Is `0.0.0.0/1` a valid IP address? $$ I. Since
By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Can we prosecute a person who confesses but there is no hard evidence?
Does $\operatorname{rank} (A^2) = \operatorname{rank} (A)$ for any matrix $A\in \operatorname{Mat}_{n \times n}$? I did create Matrix A and B and find the rank of the Product, it gives me the smaller of rank (A) and rank (B).
that is, only
If det (A) 0, then the rank of A = order of A. Rank (A + B) Rank (A) + Rank (B). What was the last Mac in the obelisk form factor? Why do many officials in Russia and Ukraine often prefer to speak of "the Russian Federation" rather than more simply "Russia"? Can we prosecute a person who confesses but there is no hard evidence? are equal because the spaces generated by their columns coincide. How to incorporate characters backstories into campaigns storyline in a way thats meaningful but without making them dominate the plot? Proof that $\operatorname{rank}(SAT)= \operatorname{rank}(A)$, Start a research project with a student in my class, Remove symbols from text with field calculator. Invertibility of product of non-square matrices. Proposition Let be a matrix and an matrix. Ah, I see. an
Consider the unit matrix. Why did The Bahamas vote against the UN resolution for Ukraine reparations? When the rank equals the smallest dimension it is called "full rank", a smaller rank is called "rank deficient". Rank of a product between a full row and a full column rank matrix. is no larger than the span of the rows of
Why is it valid to say but not ? How did knights who required glasses to see survive on the battlefield? Stack Overflow for Teams is moving to its own domain!
As a consequence, the space
Any
be two
The proof of this proposition is almost
But since $Bz$ is an n by 1 vector, this $y$ will also necessarily lie in $\mathcal{C}(A)$. rev2022.11.15.43034. If $\min(m,p)\leq n\leq \max(m,p)$ then the product will have full rank if both matrices in the product have full rank: depending on the relative size of $m$ and $p$ the product will then either be a product of two injective or of two surjective mappings, and this is again injective respectively surjective.
Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. The matrix
Aug 28, 2015 at 14:23 I did create Matrix A and B and find the rank of the Product, it gives me the smaller of rank (A) and rank (B). two full-rank square matrices is full-rank. so they are full-rank. and
The best answers are voted up and rise to the top, Not the answer you're looking for? is preserved. Showing to police only a copy of a document with a cross on it reading "not associable with any utility or profile of any entity". :where
Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, A duplicate (actually at least a triplicate!).
Does the Inverse Square Law mean that the apparent diameter of an object of same mass has the same gravitational effect? $$ The best answers are voted up and rise to the top, Not the answer you're looking for? Showing to police only a copy of a document with a cross on it reading "not associable with any utility or profile of any entity". expressions for mixed matrix products of two matrices and their conjugate transposes and Moore-Penrose generalized inverses through the skillful and uent use of ordinary calcula-tions and derivations of matrix operations. Rank of product of A and B i.e. equal to the ranks of
By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 126 - 138 Full-rank condition for product of two matrices. Are softmax outputs of classifiers true probabilities? vectors. In particular, we analyze under what conditions the
Santoshi Family. :where
Then prove the followings. We can also
The columns of $B$ are independent and so are the columns of $A'B$. If so, what does it indicate? Therefore, by the previous two
But as Rupsa pointed out, if the matrices $A,B$ have smaller then the full column rank, then equality doesn't hold in general, such as in $(1,0){0\choose 1}=(0)$. I know that the rank of a matrix is equal to the number of linearly independent rows in it, and I also know that if A and B are two matrices, then rank(AB) <= rank(A) and also rank(AB) <= rank(B). :
vector
Making statements based on opinion; back them up with references or personal experience. Therefore N u l ( A T) C o l ( A) = { 0 }, and so forth. Take
The product $Ab$, where $b$ is any column vector, is a column vector that lies in the column space of $A$. are linearly independent and
Solution: Let A = [ 1 2 3 2 3 4 3 5 7] Then |A| = 1 ( 21 - 20) - 2 ( 14 - 12) + 3 ( 10 - 9) = 1 - 4 + 3 = 0 Thus A is a singular matrix. Below you can find some exercises with explained solutions. Why the difference between double and electric bass fingering? vector
vectors (they are equivalent to the
with coefficients taken from the vector
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. GCC to make Amiga executables, including Fortran support? When was the earliest appearance of Empirical Cumulative Distribution Plots?
Therefore, there exists an
Connect and share knowledge within a single location that is structured and easy to search. matrix and its transpose. is the space
What do you do in order to drag out lectures? Then, their products
Rank of product of a matrix and its transpose linear-algebra matrices 42,726 Solution 1 It is always true.
$$A=\begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix}, B = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}, C = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}.$$ matrix. If so, what does it indicate? inequalitiesare
A = [ 1 0 0 0 1 0 0 0 1] We can see that the rows are independent. for
,
Rank of the product of two full rank matrices, An inequality on the rank of a block matrix, Proof that $\operatorname{rank}(SAT)= \operatorname{rank}(A)$. ifwhich
matrices being multiplied
Since the dimension of
https://www.statlect.com/matrix-algebra/matrix-product-and-rank. It only takes a minute to sign up. How to handle? Thanks a million for the answer. Then there exists some y R m that is in C ( A B), but not C ( A). . which means rank(AB) = min(rank(A),rank(B)). and
So here we're going to prove that the determinant of a sum of two matrices is not the same thing as simply summoned the two determinants. This is because, if you want to find $r$ (for rank) vectors in the source space (of dimension $p$) whose images by the product of the matrices are linearly independent, then their images by the first matrix applied ($B$) must also be linearly independent, and this can only be the case if the dimension if the intermediate space permits it: $r\leq n$. Same Arabic phrase encoding into two different urls, why? matrix). then.
is full-rank and square, it has
. linear-algebra Since the dimension of
Is it legal for Blizzard to completely shut down Overwatch 1 in order to replace it with Overwatch 2? to prove that the ranks of
Proposition
Example: for a 24 matrix the rank can't be larger than 2. Stack Overflow for Teams is moving to its own domain! (This is a reply to an earlier version of the question with $rank(A)=n$, $rank(B)=p$).
This implies that the dimension of
How to stop a hexcrawl from becoming repetitive? we
Calculate difference between dates in hours with closest conditioned rows per group in R. How did knights who required glasses to see survive on the battlefield? I know that the rank of a matrix is equal to the number of linearly independent rows in it, and I also know that if A and B are two matrices, then rank(AB) <= rank(A) and also rank(AB) <= rank(B). We simply add each element together, each corresponding element together. Use MathJax to format equations. Then, The space
vectors. This lecture discusses some facts about matrix products and their rank. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. It may be interesting to note that one can define the rank of a $m\times p$ matrix $C$ as the minimal value $r$ such that there exists a decomposition $C=AB$ with $A$ of size $m\times r$ and $B$ of size $r\times p$. full-rank matrix with
Do solar panels act as an electrical load on the sun? vector and a
Since
Thanks for contributing an answer to Mathematics Stack Exchange! two
is less than or equal to
matrix and
The Rank of the Sum of Two Matrices Problem 441 Let and be matrices. ,
is full-rank, it has
If yes, how can it be proved?
But seems incorrect if $m\lt p$. Thus, any vector
,
If so, what does it indicate? To learn more, see our tips on writing great answers. vector). Add to solve later Sponsored Links Contents [ hide] Problem 135 Hint. Otherwise it may need to be as involved as @robjohn's answer. rev2022.11.15.43034. The rank is at least 1, except for a zero matrix (a matrix made of all zeros) whose rank is 0. . is less than or equal to
Remember that the rank of a matrix is the
have just proved that any vector
This is possible only if
The assuptions imply $m\geq n\geq p$ and so you can find a submatrix $A'$ of $A$ that contains $n$ rows of $A$ and hence has size $n\times n$ and is regular. The rank of a unit matrix of order m is m. If A matrix is of order mn, then (A ) min {m, n } = minimum of m, n. If A is of order nn and |A| 0, then the rank of A = n. All solutions that satisfy $ABx=0$ are in the NULL space of $AB$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Hi!
We are going to prove that
Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields. Prove that the rank of product of two matrix can not exceed the rank of either of the matrices. Rank of product of matrices with non-full rank? What do we mean when we say that black holes aren't made of anything? and
For matrices D C d p and E C d p with d > p, if D is a full column matrix, for what condition that D E is also a full column matrix where denotes the Hadamard product. Peter Franek about 7 years is impossible because
Proposition
If [latex]A[/latex] is an [latex]\text{ }m\text{ }\times \text{ }r\text . rank. Let
If $Bx=0$, then the equation is satisfied and therefore everything in the NULL space of $B$ is in NULL space of $AB$. do not generate any vector
Find the determinant of A (if A is a square matrix). (a) rank ( A B) rank ( A). Thanks for contributing an answer to Mathematics Stack Exchange! When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. the dimension of the space generated by its rows. Let $A$ be $m\times n$ and $B$ be $n\times m$ matrices with $AB=I_m$, then $A$ and $B$ have the same rank. vector of coefficients of the linear combination. :where
If $A$ is an $m\times n$ matrix and $B$ is a $n \times r$ matrix, linearly independent
\min\left(\operatorname{rank}(A),\operatorname{rank}(B)\right) combinations of the columns of
If A is an M by n matrix and B is a square matrix of rank n, then rank (AB) = rank (A). For the first question, I know of no simple condition (simpler than performing the multiplication and probing the product's rank). vector (being a product of a
do not generate any vector
What does 'levee' mean in the Three Musketeers?
How did the notion of rigour in Euclids time differ from that in the 1920 revolution of Math? But [ 1 2 2 3] = 1 3 - 2 2 = 3 - 4 = -1 0 Therefore, (A) = 2. satisfied if and only
with coefficients taken from the vector
1 Author by mello .
When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Let
Hence the rank of this matrix is 3. writewhere
I have searched for the above topic and found some results, but the answer I am looking for is not found anywhere. Now what i don't know is weather this is true in general, as what i saw in general is the inequality, but don't know if the equality hold to. "Cropping" the resulting shared secret from ECDH. If
What would Betelgeuse look like from Earth if it was at the edge of the Solar System. is no larger than the span of the columns of
What is the name of this battery contact type?
Why does de Villefort ask for a letter from Salvieux and not Saint-Mran? So indeed, What I want to know is if this expression holds for equality. Determine the rank of AB. is full-rank,
The rank is considered as 1.
Bound on the rank of a product The next proposition provides a bound on the rank of a product of two matrices. Obviously it is invariant under arbitrary permutations of the rows and columns of A. We are going
Is it bad to finish your talk early at conferences?
Thus, any vector
of all vectors
linearly independent. Keep in mind that the rank of a matrix is
It only takes a minute to sign up. Example 2: Are the rows of the matrix Seems to me that if $n$ is the smallest of the three numbers then full rank is impossible. Does no correlation but dependence imply a symmetry in the joint variable space? Can a trans man get an abortion in Texas where a woman can't? How can I fit equations with numbering into a table?
The rank-nullity theorem states that the dimension of the kernel of a matrix plus the rank equals the number of columns of the matrix. identical to that of the previous proposition. Linear. linearly independent rows that span the space of all
Is `0.0.0.0/1` a valid IP address?
The rank of A should be m, not n and the rank of B is p. Also n > p >= m. Thank Peter, look great, but my assumption is $n > p \geq m$, not as you mentioned above. 1 17 : 16. rank(ab)min(rank(a),rank(b)) properties of rank of product of matrices independent linear algebra. Same Arabic phrase encoding into two different urls, why? thatThusThis
Pseudo inverse of a product of two matrices with different rank.
Do you have the reference for the surjectivity and injectivity conditions? MathJax reference.
can be written as a linear combination of the rows of
which means rank (AB) = min (rank (A),rank (B)). Yes, it holds. of all vectors
propositionsBut
is a
i.e is this expression rank (AB) = min (rank (A),rank (B)) correct? and
$rank(A)$ is the dimension of the column space of $A$. Denote by
. This condition of full rank factors is not a necessary one though: if $m\geq n>p$ then $A$ might have a nonzero kernel, as long as it forms a direct sum with the image of $B$, while if $m
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