Understanding Matrices | Half 2: Matrix-Matrix Multiplication

Within the first story [1] of this sequence, we’ve got:

• Addressed multiplication of a matrix by a vector,
• Launched the idea of X-diagram for a given matrix,
• Noticed conduct of a number of particular matrices, when being multiplied by a vector.

Within the present 2nd story, we’ll grasp the bodily that means of matrix-matrix multiplication, perceive why multiplication is just not a symmetrical operation (i.e., why “A*B ≠ B*A“), and eventually, we’ll see how a number of particular matrices behave when being multiplied over one another.

So let’s begin, and we’ll do it by recalling the definitions that I exploit all through this sequence:

• Matrices are denoted with uppercase (like ‘A‘ and ‘B‘), whereas vectors and scalars are denoted with lowercase (like ‘x‘, ‘y‘ or ‘m‘, ‘n‘).
• |x| – is the size of vector ‘x‘,
• rows(A) – variety of rows of matrix ‘A‘,
• columns(A) – variety of columns of matrix ‘A‘.

The idea of multiplying matrices

Multiplication of two matrices “A” and “B” might be the most typical operation in matrix evaluation. A identified reality is that “A” and “B” could be multiplied provided that “columns(A) = rows(B)”. On the similar time, “A” can have any variety of rows, and “B” can have any variety of columns. Cells of the product matrix “C = A*B” are calculated by the next formulation:

[
begin{equation*}
c_{i,j} = sum_{k=1}^{p} a_{i,k}*b_{k,j}
end{equation*}
]

the place “p = columns(A) = rows(B)”. The consequence matrix “C” could have the scale:

rows(C) = rows(A),
columns(C) = columns(B).

Performing upon the multiplication formulation, when calculating “A*B” we must always scan i-th row of “A” in parallel to scanning j-th column of “B“, and after summing up all of the merchandise “a_i,okay*b_okay,j” we could have the worth of “c_i,j“.

One other well-known reality is that matrix multiplication is just not a symmetrical operation, i.e., “A*B ≠ B*A“. With out going into particulars, we are able to already see that when multiplying 2 rectangular matrices:

For newbies, the truth that matrix multiplication is just not a symmetrical operation typically appears unusual, as multiplication outlined for nearly every other object is a symmetrical operation. One other reality that’s typically unclear is why matrix multiplication is carried out by such an odd formulation.

On this story, I’m going to provide my solutions to each of those questions, and never solely to them…

Derivation of the matrices multiplication formulation

Multiplying “A*B” ought to produce such a matrix ‘C‘, that:

y = C*x = (A*B)*x = A*(B*x).

In different phrases, multiplying any vector ‘x‘ by the product matrix “C=A*B” ought to end in the identical vector ‘y‘, which we’ll obtain if at first multiplying ‘B‘ by ‘x‘, after which multiplying ‘A‘ by that intermediate consequence.

This already explains why in “C=A*B“, the situation that “columns(A) = rows(B)” needs to be stored. That’s due to the size of the intermediate vector. Let’s denote it as ‘t‘:

t = B*x,
y = C*x = (A*B)*x = A*(B*x) = A*t.

Clearly, as “t = B*x“, we’ll obtain a vector ‘t‘ of size “|t| = rows(B)”. However later, matrix ‘A‘ goes to be multiplied by ‘t‘, which requires ‘t‘ to have the size “|t| = columns(A)”. From these 2 information, we are able to already determine that:

rows(B) = |t| = columns(A), or
rows(B) = columns(A).

Within the first story [1] of this sequence, we’ve got realized the “X-way interpretation” of matrix-vector multiplication “A*x“. Contemplating that for “y = (A*B)x“, vector ‘x‘ goes at first via the transformation of matrix ‘B‘, after which it continues via the transformation of matrix ‘A‘, we are able to broaden the idea of “X-way interpretation” and current matrix-matrix multiplication “A*B” as 2 adjoining X-diagrams:

Now, what ought to a sure cell “c_i,j” of matrix ‘C‘ be equal to? From half 1 – “matrix-vector multiplication” [1], we do not forget that the bodily that means of “c_i,j” is – how a lot the enter worth ‘x_j‘ impacts the output worth ‘y_i‘. Contemplating the image above, let’s see how some enter worth ‘x_j‘ can have an effect on another output worth ‘y_i‘. It will probably have an effect on via the intermediate worth ‘t₁‘, i.e., via arrows “a_i,1” and “b_1,j“. Additionally, the love can happen via the intermediate worth ‘t₂‘, i.e., via arrows “a_i,2” and “b_2,j“. Typically, the love of ‘x_j‘ on ‘y_i‘ can happen via any worth ‘t_okay‘ of the intermediate vector ‘t‘, i.e., via arrows “a_i,okay” and “b_okay,j“.

So there are ‘p‘ attainable methods through which the worth ‘x_j‘ influences ‘y_i‘, the place ‘p‘ is the size of the intermediate vector: “p = |t| = |B*x|”. The influences are:

[begin{equation*}
begin{matrix}
a_{i,1}*b_{1,j},
a_{i,2}*b_{2,j},
a_{i,3}*b_{3,j},
dots
a_{i,p}*b_{p,j}
end{matrix}
end{equation*}]

All these ‘p‘ influences are unbiased of one another, which is why within the formulation of matrices multiplication they take part as a sum:

[begin{equation*}
c_{i,j} =
a_{i,1}*b_{1,j} + a_{i,2}*b_{2,j} + dots + a_{i,p}*b_{p,j} =
sum_{k=1}^{p} a_{i,k}*b_{k,j}
end{equation*}]

That is my visible clarification of the matrix-matrix multiplication formulation. By the way in which, decoding “A*B” as a concatenation of X-diagrams of “A” and “B” explicitly exhibits why the situation “columns(A) = rows(B)” needs to be held. That’s easy, as a result of in any other case it won’t be attainable to concatenate the 2 X-diagrams:

Why is it that “A*B ≠ B*A”

Decoding matrix multiplication “A*B” as a concatenation of X-diagrams of “A” and “B” additionally explains why multiplication is just not symmetrical for matrices, i.e., why “A*B ≠ B*A“. Let me present that on two sure matrices:

[begin{equation*}
A =
begin{bmatrix}
0 & 0 & 0 & 0
0 & 0 & 0 & 0
a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4}
a_{4,1} & a_{4,2} & a_{4,3} & a_{4,4}
end{bmatrix}
, B =
begin{bmatrix}
b_{1,1} & b_{1,2} & 0 & 0
b_{2,1} & b_{2,2} & 0 & 0
b_{3,1} & b_{3,2} & 0 & 0
b_{4,1} & b_{4,2} & 0 & 0
end{bmatrix}
end{equation*}]

Right here, matrix ‘A‘ has its higher half stuffed with zeroes, whereas ‘B‘ has zeroes on its proper half. Corresponding X-diagrams are:

What’s going to occur if attempting to multiply “A*B“? Then A’s X-diagram needs to be positioned to the left of B’s X-diagram.

Having such a placement, we see that enter values ‘x₁‘ and ‘x₂‘ can have an effect on each output values ‘y₃‘ and ‘y₄‘. Significantly, which means that the product matrix “A*B” is non-zero.

[
begin{equation*}
A*B =
begin{bmatrix}
0 & 0 & 0 & 0
0 & 0 & 0 & 0
c_{3,1} & c_{3,2} & 0 & 0
c_{4,1} & c_{4,2} & 0 & 0
end{bmatrix}
end{equation*}
]

Now, what is going to occur if we attempt to multiply these two matrices within the reverse order? For presenting the product “B*A“, B’s X-diagram needs to be drawn to the left of A’s diagram:

We see that now there isn’t a linked path, by which any enter worth “x_j” can have an effect on any output worth “y_i“. In different phrases, within the product matrix “B*A” there isn’t a affection in any respect, and it’s really a zero-matrix.

[begin{equation*}
B*A =
begin{bmatrix}
0 & 0 & 0 & 0
0 & 0 & 0 & 0
0 & 0 & 0 & 0
0 & 0 & 0 & 0
end{bmatrix}
end{equation*}]

This instance clearly illustrates why order is vital for matrix-matrix multiplication. After all, many different examples can be found out.

Multiplying chain of matrices

X-diagrams can be concatenated once we multiply 3 or extra matrices. For example, for the case of:

G = A*B*C,

we are able to draw the concatenation within the following means:

Right here we now have 2 intermediate vectors:

t = C*x, and
s = (B*C)*x = B*(C*x) = B*t

whereas the consequence vector is:

y = (A*B*C)*x = A*(B*(C*x)) = A*(B*t) = A*s.

The variety of attainable methods through which some enter worth “x_j” can have an effect on some output worth “y_i” grows right here by an order of magnitude.

Extra exactly, the affect of sure “x_j” over “y_i” can come via any merchandise of the primary intermediate stack “t“, and any merchandise of the second intermediate stack “s“. So the variety of methods of affect turns into “|t|*|s|”, and the formulation for “g_i,j” turns into:

[begin{equation*}
g_{i,j} = sum_{v=1}^ sum_{u=1}^ a_{i,v}*b_{v,u}*c_{u,j}
end{equation*}]

Multiplying matrices of particular varieties

We will already visually interpret matrix-matrix multiplication. Within the first story of this sequence [1], we additionally realized about a number of particular kinds of matrices – the dimensions matrix, shift matrix, permutation matrix, and others. So let’s check out how multiplication works for these kinds of matrices.

Multiplication of scale matrices

A scale matrix has non-zero values solely on its diagonal:

From idea, we all know that multiplying two scale matrices leads to one other scale matrix. Why is it that means? Let’s concatenate X-diagrams of two scale matrices:

The concatenation X-diagram clearly exhibits that any enter merchandise “x_i” can nonetheless have an effect on solely the corresponding output merchandise “y_i“. It has no means of influencing every other output merchandise. Subsequently, the consequence construction behaves the identical means as another scale matrix.

Multiplication of shift matrices

A shift matrix is one which, when multiplied over some enter vector ‘x‘, shifts upwards or downwards values of ‘x‘ by some ‘okay‘ positions, filling the emptied slots with zeroes. To realize that, a shift matrix ‘V‘ will need to have 1(s) on a line parallel to its predominant diagonal, and 0(s) in any respect different cells.

The idea says that multiplying 2 shift matrices ‘V1‘ and ‘V2‘ leads to one other shift matrix. Interpretation with X-diagrams offers a transparent clarification of that. Multiplying the shift matrices ‘V1‘ and ‘V2‘ corresponds to concatenating their X-diagrams:

We see that if shift matrix ‘V1‘ shifts values of its enter vector by ‘k1‘ positions upwards, and shift matrix ‘V2‘ shifts values of the enter vector by ‘k2‘ positions upwards, then the outcomes matrix “V3 = V1*V2” will shift values of the enter vector by ‘k1+k2‘ positions upwards, which signifies that “V3” can be a shift matrix.

Multiplication of permutation matrices

A permutation matrix is one which, when multiplied by an enter vector ‘x‘, rearranges the order of values in ‘x‘. To behave like that, the NxN-sized permutation matrix ‘P‘ should fulfill the next standards:

• it ought to have N 1(s),
• no two 1(s) needs to be on the identical row or the identical column,
• all remaining cells needs to be 0(s).

Upon idea, multiplying 2 permutation matrices ‘P1‘ and ‘P2‘ leads to one other permutation matrix ‘P3‘. Whereas the explanation for this may not be clear sufficient if taking a look at matrix multiplication within the abnormal means (as scanning rows of ‘P1‘ and columns of ‘P2‘), it turns into a lot clearer if taking a look at it via the interpretation of X-diagrams. Multiplying “P1*P2” is identical as concatenating X-diagrams of ‘P1‘ and ‘P2‘.

We see that each enter worth ‘x_j‘ of the proper stack nonetheless has just one path for reaching another place ‘y_i‘ on the left stack. So “P1*P2” nonetheless acts as a rearrangement of all values of the enter vector ‘x‘, in different phrases, “P1*P2” can be a permutation matrix.

Multiplication of triangular matrices

A triangular matrix has all zeroes both above or beneath its predominant diagonal. Right here, let’s think about upper-triangular matrices, the place zeroes are beneath the primary diagonal. The case of lower-triangular matrices is comparable.

The truth that non-zero values of ‘B‘ are both on its predominant diagonal or above, makes all of the arrows of its X-diagram both horizontal or directed upwards. This, in flip, signifies that any enter worth ‘x_j‘ of the proper stack can have an effect on solely these output values ‘y_i‘ of the left stack, which have a lesser or equal index (i.e., “i ≤ j“). That is without doubt one of the properties of an upper-triangular matrix.

In accordance with idea, multiplying two upper-triangular matrices leads to one other upper-triangular matrix. And right here too, interpretation with X-diagrams supplies a transparent clarification of that reality. Multiplying two upper-triangular matrices ‘A‘ and ‘B‘ is identical as concatenating their X-diagrams:

We see that placing two X-diagrams of triangular matrices ‘A‘ and ‘B‘ close to one another leads to such a diagram, the place each enter worth ‘x_j‘ of the proper stack nonetheless can have an effect on solely these output values ‘y_i‘ of the left stack, that are both on its degree or above it (in different phrases, “i ≤ j“). Which means that the product “A*B” additionally behaves like an upper-triangular matrix; thus, it will need to have zeroes beneath its predominant diagonal.

Conclusion

Within the present 2nd story of this sequence, we noticed how matrix-matrix multiplication could be introduced visually, with the assistance of so-called “X-diagrams”. We’ve realized that doing multiplication “C = A*B” is identical as concatenating X-diagrams of these two matrices. This technique clearly illustrates numerous properties of matrix multiplications, like why it isn’t a symmetrical operation (“A*B ≠ B*A“), in addition to explains the formulation:

[begin{equation*}
c_{i,j} = sum_{k=1}^{p} a_{i,k}*b_{k,j}
end{equation*}]

We’ve additionally noticed why multiplication behaves in sure methods when operands are matrices of particular varieties (scale, shift, permutation, and triangular matrices).

I hope you loved studying this story!

Within the coming story, we’ll tackle how matrix transposition “A^T” could be interpreted with X-diagrams, and what we are able to achieve from such interpretation, so subscribe to my web page to not miss the updates!

My gratitude to:
– Roza Galstyan, for cautious evaluation of the draft ( https://www.linkedin.com/in/roza-galstyan-a54a8b352 )
– Asya Papyan, for the exact design of all of the used illustrations ( https://www.behance.web/asyapapyan ).

In case you loved studying this story, be happy to observe me on LinkedIn, the place, amongst different issues, I may also submit updates ( https://www.linkedin.com/in/tigran-hayrapetyan-cs/ ).

All used photos, except in any other case famous, are designed by request of the writer.

References

[1] – Understanding matrices | Half 1: matrix-vector multiplication : https://towardsdatascience.com/understanding-matrices-part-1-matrix-vector-multiplication/