Review of similarity transformation and Singular Value Decomposition Nasser M. Abbasi Applied Mathematics Department, California State University, Fullerton. July 8 2007 Compiled on May 19, 2020 at 2:03am Contents 1 Similarity transformation 2 1.1 Derivation of similarity transformation based on algebraic method .......... 2 1.2 Derivation of similarity transformation based on geometric method ......... 3 1.2.1 Finding matrix representation of linear transformation ........... 5 1.2.2 Finding matrix representation of change of basis ................ 6 1.2.3 Examples of similarity transformation ...................... 8 1.2.4 How to ๏ฌnd the best basis to simplify the representation of ? ........ 11 1.3 Summary of similarity transformation ........................... 12 2 Singular value decomposition (SVD) 12 2.1 What is right and left eigenvectors? ............................ 12 2.2 SVD .............................................. 12 3 Conclusion 14 4 References 15 1
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Review of similarity transformation and Singular Value ...โฌยฆย ยท 2 1 Similarity transformation A similarity transformation is ๐ต=๐โ1๐ด๐ Where ๐ต,๐ด,๐are square
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Review of similarity transformation and Singular ValueDecomposition
Nasser M. Abbasi
Applied Mathematics Department, California State University, Fullerton. July 8 2007 Compiled on May 19, 2020 at
2:03am
Contents
1 Similarity transformation 21.1 Derivation of similarity transformation based on algebraic method . . . . . . . . . . 21.2 Derivation of similarity transformation based on geometric method . . . . . . . . . 3
1.2.1 Finding matrix representation of linear transformation ๐ . . . . . . . . . . . 51.2.2 Finding matrix representation of change of basis . . . . . . . . . . . . . . . . 61.2.3 Examples of similarity transformation . . . . . . . . . . . . . . . . . . . . . . 81.2.4 How to find the best basis to simplify the representation of ๐? . . . . . . . . 11
A similarity transformation is๐ต = ๐โ1๐ด๐
Where ๐ต,๐ด,๐ are square matrices. The goal of similarity transformation is to find a ๐ต matrix whichhas a simpler form than ๐ด so that we can use ๐ต in place of ๐ด to ease some computational work.Lets set our goal in having ๐ต be a diagonal matrix (a general diagonal form is called block diagonalor Jordan form, but here we are just looking at the case of ๐ต being a diagonal matrix).
The question becomes: Given ๐ด, can we find ๐ such that ๐โ1๐ด๐ is diagonal?
The standard method to show the above is via an algebraic method, which we show first. Next weshow a geometric method that explains similarity transformation geometrically.
1.1 Derivation of similarity transformation based on algebraic method
Starting with๐ต = ๐โ1๐ด๐
Our goal is to find a real matrix ๐ต such that it is diagonal. From the above, by pre multiplyingeach side by ๐ we obtain
๐ด๐ = ๐๐ต
Now, since our goal is to make ๐ต diagonal, let us select the eigenvalues of ๐ด to be the diagonal of๐ต. Now we write the above in expanded form as follows
The above shows that if we take the diagonal elements of ๐ต to be the eigenvalues of ๐ด then the ๐matrix is the (right) eigenvectors of ๐ด.
Hence, if we want the ๐ต matrix to be diagonal and real, this means the ๐ด matrix itself must havesome conditions put on it. Specifically, ๐ด eigenvalues must be all distinct and real. This happenswhen ๐ด is real and symmetric. Therefore, we have the general result that given a matrix ๐ด whichhas distinct and real eigenvalues, only then can we find a similar matrix to it which is real anddiagonal, and these diagonal values are the eigenvalues of ๐ด.
3
1.2 Derivation of similarity transformation based on geometric method
It turns out that this derivation is more complicated than the algebraic approach. It involves achange of basis matrix (which will be our๐ matrix) and the representation of linear transformation๐ as a matrix under some basis (which will be our ๐ด matrix). Let us start from the beginning.
Given the vector ๐ in โ๐, it can have many di๏ฟฝerent representations (or coordinates) depending onwhich basis are used. There are an infinite number of basis that span โ๐, hence there are infiniterepresentations for the same vector. Consider for example โ2. The standard basis vectors areโงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ10
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ, but another basis are
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ, and yet another are
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ0โ1
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ, and so on.
basis v
basis v
basis vChange of basis
In โ๐ any ๐ linearly independent vectors can be used as basis. Let ๐๐ฃ be the vector representationin w.r.t. basis ๐ฃ, and let ๐๐ฃโฒ be the vector representation w.r.t. basis ๐ฃโฒ.
basis v
basisv
Change of basis
Same vector X has different representation depending on which basis are used
xv x,yxv x,y
Mv v
basis v
An important problem in linear algebra is to obtain ๐๐ฃ given ๐๐ฃโฒ and the change of basis matrix[๐]๐ฃโฒโ๐ฃ.
This requires finding a matrix representation for the change of basis. Once the [๐]๐ฃโฒโ๐ฃ matrix isfound, we can write
๐๐ฃ = [๐]๐ฃโฒโ๐ฃ ๐๐ฃโฒ (1)
Where [๐]๐ฃโฒโ๐ฃ is the change of basis matrix which when applied to ๐๐ฃโฒ returns the coordinates ofthe vector w.r.t. basis ๐ฃ.
From (1) we see that given ๐๐ฃ, then to obtain ๐๐ฃโฒ we write
Another important problem is that of linear transformation ๐, where now we apply some transfor-mation on the whole space and we want to find what happens to the space coordinates (all theposition vectors) due to this transformation.
Consider for example ๐ which is a rotation in โ2 by some angle ๐. Here, every position vector in thespace is a๏ฟฝected by this rotation but the basis remain ๏ฟฝxed, and we want to find the new coordinatesof each point in space.
Let ๐๐ฃ be the position vector before applying the linear transformation, and let ๐๐ฃ be the newposition vector. Notice that both vectors are written with respect to the same basis ๐ฃ. Similar tochange basis, we want a way to find ๐๐ฃ from ๐๐ฃ, hence we need a way to represent this lineartransformation by a matrix, say [๐]๐ฃ with respect to the basis ๐ฃ and so we could write the following
๐๐ฃ = [๐]๐ฃ ๐๐ฃ (2)
4
basis v
Linear Transformation T
Same basis v but T causes change in vector representation
basis vT
v
yv
xv xv
Assume the basis is changed to ๐ฃโฒ. Let the representation of the same linear transformation ๐ w.r.t.basis ๐ฃโฒ be called [๐]๐ฃโฒ, so we write
๐๐ฃโฒ = [๐]๐ฃโฒ ๐๐ฃโฒ (2A)
basis vChange basis v->vโ
Combining change of basis and linear transformation
xv
basis v
xv
basis v
xv
yv T
v xv
Linear Transformatio
n
Tv
Hence when the basis changes from ๐ฃ to ๐ฃโฒ, ๐ representation changes from [๐]๐ฃ to [๐]๐ฃโฒ .
Now assume we are given some linear transformation ๐ and its representation [๐]๐ฃ w.r.t. basis ๐ฃ,how could we find ๐ representation [๐]๐ฃโฒ w.r.t. to new basis ๐ฃโฒ?
From (2) we have๐๐ฃ = [๐]๐ฃ ๐๐ฃ
But from (1) ๐๐ฃ = [๐]๐ฃโฒโ๐ฃ ๐๐ฃโฒ, hence the above becomes
Notice the di๏ฟฝerence between change of basis and linear transformation. In change of basis, thevector ๐ remained fixed in space, but the basis changed, and we want to find the coordinates of thevector w.r.t the new basis. With linear transformation, the vector itself is changed from ๐๐ฃ to ๐๐ฃ, butboth vectors are expressed w.r.t. the same basis.
Equation (3) allows us to obtain a new matrix representation of the linear transformation ๐ byexpressing the same ๐ w.r.t. to di๏ฟฝerent basis. Hence if we are given a representation of ๐ which isnot the most optimal representation, we can, by change of basis, obtain a di๏ฟฝerent representationof the same ๐ by using (3). The most optimal matrix representation for linear transformation is adiagonal matrix.
5
Equation (3) is called a similarity transformation. To make it easier to compare (3) above withwhat we wrote in the previous section when we derived the above using an algebraic approach, welet [๐]๐ฃโฒ = ๐ต, [๐]๐ฃ = ๐ด, hence (3) is
๐ต = ๐โ1๐ด๐
The matrix [๐]๐ฃโฒ is similar to [๐]๐ฃ. (i.e. ๐ต is similar to ๐ด). Both matrices represent the same lineartransformation applied on the space. We will show below some examples of how to find [๐]๐ฃโฒ given[๐]๐ฃ and [๐]. But first we show how to obtain matrix representation of ๐.
1.2.1 Finding matrix representation of linear transformation ๐
Given a vector ๐ โ โ๐ and some linear transformation (or a linear operator) ๐ that acts on thevector ๐ transforming this vector into another vector ๐ โ โ๐ according to some prescribed manner๐ โถ ๐ โ ๐. Examples of such linear transformation are rotation, elongation, compression, andreflection, and any combination of these operations, but it can not include the translation of thevector, since translation is not linear.
The question to answer here is how to write down a representation of this linear transformation? Itturns out that ๐ can be represented by a matrix of size ๐ ร ๐, and the actual numbers that go intothe matrix, or the shape of the matrix, will depend on which basis in โ๐ we choose to represent ๐.
We would like to pick some basis so that the final shape of the matrix is the most simple shape.
Let us pick a set of basis ๐ฃ = {๐1, ๐2,โฏ , ๐๐} that span โ๐ hence
We see from above that the new transformed vector ๐๐๐ฃ has the same coordinates as ๐๐ฃ if we view๐๐๐ as the new basis.
Now ๐๐๐ itself is an application of the linear transformation but now it is being done on each basisvector ๐๐. Hence it will cause each specific basis vector to transform in some manner to a new vector.Whatever this new vector is, we must be able to represent it in terms of the same basis vectors{๐1, ๐2,โฏ , ๐๐} , therefore, we write
Hence, since ๐๐๐ฃ = ๐1๐๐1 +๐2๐๐2 +โฏ+๐๐๐๐๐ from (1), then by comparing coe๏ฟฝcients of each basisvector ๐๐ we obtain
Let us call the matrix that represents ๐ under basis ๐ฃ as [๐]๐ฃ .
We see from the above that the ๐๐กโ column of [๐]๐ฃ contain the coordinates of the vector ๐๐๐. Thisgives us a quick way to construct [๐]๐ฃ: Apply ๐ to each basis vector ๐๐, and take the resulting vectorand place it in the ๐๐กโ column of [๐]๐ฃ.
We now see that [๐]๐ฃ will have a di๏ฟฝerent numerical values if the basis used to span โ๐ are di๏ฟฝerentfrom the ones used to obtain the above [๐]๐ฃ .
Let use pick some new basis, say ๐ฃโฒ = ๏ฟฝ๐โฒ1, ๐โฒ2,โฏ , ๐โฒ๐๏ฟฝ. Let the new representation of ๐ now be thematrix [๐]๐ฃโฒ, then the ๐๐กโ column of [๐]๐ฃโฒ is found from applying ๐ on the new basis ๐โฒ๐
๐๐โฒ๐ = ๐๐๐๐โฒ๐
Where now ๐๐๐ is the ๐๐กโ coordinates of the the vector ๐๐โฒ๐ which will be di๏ฟฝerent from ๐๐๐ since in
one case we used the basis set ๐ฃโฒ = ๏ฟฝ๐โฒ๐๏ฟฝ and in the other case we used the basis set ๐ฃ = {๐๐}. Hencewe see that [๐]๐ฃโฒ will numerically be di๏ฟฝerent depending on the basis used to span the space eventhough the linear transformation ๐ itself did not change.
1.2.2 Finding matrix representation of change of basis
Now we show how to determine [๐]๐ฃโ๐ฃโฒ, the matrix which when applied to a vector ๐๐ฃ will resultin the vector ๐๐ฃโฒ.
Given a vector ๐, it is represented w.r.t. basis ๐ฃ = {๐1, ๐2,โฏ , ๐๐} as
The above gives a relation between the coordinates of the vector ๐ w.r.t. basis ๐ฃ (these are the ๐๐coordinates) to the coordinates of the same vector w.r.t. to basis ๐ฃโฒ (these are the coordinates ๐๐).The mapping between these coordinates is the matrix shown above which we call the [๐] matrix.Since the above matrix returns the coordinates of the vector w.r.t. ๐ฃ when it is multiplied by thecoordinates of the vector w.r.t. basis ๐ฃโฒ, we write it as [๐]๐ฃโฒโ๐ฃ to make it clear from which basis towhich basis is the conversion taking place.
Looking at (2) and (3) above, we see that column ๐ in [๐]๐ฃโฒโ๐ฃ is the coordinates of the ๐โฒ๐ w.r.t. basis
๐ฃ.
Hence the above gives a method to generate the change of basis matrix [๐]๐ฃโฒโ๐ฃ. Simply representeach basis in ๐ฃโฒ w.r.t. to basis ๐ฃ. Take the result of each such representation and write it as a columnin [๐]๐ฃโฒโ๐ฃ. We now show few examples to illustrate this.
Example showing how to ๏ฟฝnd change of basis matrix
Given the โ2 basis ๐ฃ = {๐1, ๐2} =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ10
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ, and new basis ๐ฃโฒ = ๏ฟฝ๐โฒ1, ๐โฒ2๏ฟฝ =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃโ11
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ, find the change
of basis matrix [๐]๐ฃโ๐ฃโฒ
8
Column 1 of [๐]๐ฃโ๐ฃโฒ is the coordinates of ๐1 w.r.t. basis ๐ฃโฒ. i.e.
๐1 = ๐11๐โฒ1 + ๐21๐โฒ2 (4)
and column 2 of [๐]๐ฃโ๐ฃโฒ is the coordinates of ๐2 w.r.t. basis ๐ฃโฒ. i.e.
๐2 = ๐12๐โฒ1 + ๐22๐โฒ2 (5)
But (4) is โกโขโขโขโขโฃ10
โคโฅโฅโฅโฅโฆ = ๐11
โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ + ๐21
โกโขโขโขโขโฃโ11
โคโฅโฅโฅโฅโฆ
Hence 1 = ๐11 โ ๐21 and 0 = ๐11 + ๐21, solving, we obtain ๐11 =12 , ๐21 =
โ12 and (5) is
โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ = ๐12
โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ + ๐22
โกโขโขโขโขโฃโ11
โคโฅโฅโฅโฅโฆ
Hence 0 = ๐12 โ ๐22 and 1 = ๐12 + ๐22, solving we obtain ๐12 =12 , ๐22 =
12 , hence
[๐]๐ฃโ๐ฃโฒ =โกโขโขโขโขโฃ12
12
โ12
12
โคโฅโฅโฅโฅโฆ
Now we can use the above change of basis matrix to find coordinates of the vector under ๐ฃโฒ given
its coordinates under ๐ฃ. For example, given ๐๐ฃ =โกโขโขโขโขโฃ105
โคโฅโฅโฅโฅโฆ, then
๐๐ฃโฒ =โกโขโขโขโขโฃ12
12
โ12
12
โคโฅโฅโฅโฅโฆ
โกโขโขโขโขโฃ105
โคโฅโฅโฅโฅโฆ =
โกโขโขโขโขโฃ7.5โ2.5
โคโฅโฅโฅโฅโฆ
1.2.3 Examples of similarity transformation
Example 1 This example from Strang book (Linear Algebra and its applications, 4th edition),but shown here with little more details.
Consider โ2, and let ๐ be the projection onto a line at angle ๐ = 1350. Let the first basis ๐ฃ =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ10
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ
and let the second basis ๐ฃโฒ =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ1โ1
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ.
basis v basis v
e1
e2
e1
e2
1
0
0
1
Te1 0.5
0.5
1350
Te2 0.5
0.5
Tv
0.5 0.5
0.5 0.5
Tv
1 0
0 0
Change of basis
The first step is to find [๐]๐ฃโฒโ๐ฃ. The first column of [๐]๐ฃโฒโ๐ฃ is found by writing ๐โฒ1w.r.t. basis ๐ฃ, andthe second column of [๐]๐ฃโฒโ๐ฃ is found by writing ๐โฒ2w.r.t. basis ๐ฃ, Hence
Now we need to find [๐]๐ฃ the representation of ๐ w.r.t. basis ๐ฃ. The first column of [๐]๐ฃ is the newcoordinates of the basis ๐ฃ1 after applying ๐ onto it. but
๐๐1 = ๐โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
=โกโขโขโขโขโฃ0.5โ0.5
โคโฅโฅโฅโฅโฆ
and the second column of [๐]๐ฃ is the new coordinates of the basis ๐ฃ2 after applying ๐ onto it. but
Hence we see that the linear transformation ๐ is represented as
โกโขโขโขโขโฃ1 00 0
โคโฅโฅโฅโฅโฆ w.r.t. basis ๐ฃโฒ, while it is
represented as
โกโขโขโขโขโฃ0.5 โ0.5โ0.5 0.5
โคโฅโฅโฅโฅโฆw.r.t. basis ๐ฃโฒ. Therefore, it will be better to perform all calculations
involving this linear transformation under basis ๐ฃโฒ instead of basis ๐ฃ
Example 2
10
basis v
basis v
e1
e2
e1
e2
1
0
0
1
Change of basis
4
Te1 cos
sinTe2
sin
cos
Tv
cos sin
sin cos
Tv
1
2 1
2
1
2
1
2
1
1
1
1
Tv
1
2 1
2
1
2
1
2
Change of basis selected did not result in simplification of representation of T
Consider โ2, and let ๐ be a rotation of space by ๐ = ๐4 . Let the first basis ๐ฃ =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ10
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญand let the
second basis be ๐ฃโฒ =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ11
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃโ11
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญ.
The first step is to find [๐]๐ฃโฒโ๐ฃ. The first column of [๐]๐ฃโฒโ๐ฃ is found by writing ๐โฒ1w.r.t. basis ๐ฃ, andthe second column of [๐]๐ฃโฒโ๐ฃ is found by writing ๐โฒ2w.r.t. basis ๐ฃ, Hence
Now we need to find [๐]๐ฃ the representation of ๐ w.r.t. basis ๐ฃ. The first column of [๐]๐ฃ is the newcoordinates of the basis ๐ฃ1 after applying ๐ onto it. but
๐๐1 = ๐โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
=โกโขโขโขโขโฃcos๐sin๐
โคโฅโฅโฅโฅโฆ
and the second column of [๐]๐ฃ is the new coordinates of the basis ๐ฃ2 after applying ๐ onto it. but
Hence we see that the linear transformation ๐ is represented as
โกโขโขโขโขโขโขโฃ
1
โ2โ 1
โ21
โ21
โ2
โคโฅโฅโฅโฅโฅโฅโฆ w.r.t. basis ๐ฃ
โฒ, while it is
represented as
โกโขโขโขโขโขโขโฃ
1
โ2โ 1
โ21
โ21
โ2
โคโฅโฅโฅโฅโฅโฅโฆw.r.t. basis ๐ฃ
โฒ. Therefore the change of basis selected above did not result
in making the representation of ๐ any simpler (in fact there was no change in the representation).This means we need to find a more direct method of finding the basis under which ๐ has thesimplest representation. Clearly we canโt just keep trying di๏ฟฝerent basis to find if ๐ has simplerrepresentation under the new basis.
1.2.4 How to ๏ฟฝnd the best basis to simplify the representation of ๐?
Our goal of simpler representation of ๐ is that of a diagonal matrix or as close as possible to beingdiagonal matrix (i.e. Jordan form). Given [๐]๐ฃ and an infinite number of basis ๐ฃโฒ that we couldselect to represent ๐ under in the hope we can find a new representation [๐]๐ฃโฒ such that it is simplerthan [๐]๐ฃ, we now ask, how does one go about finding such basis?
It turns out the if we select the eigenvectors of [๐]๐ฃ as the columns of [๐]๐ฃโฒโ๐ฃ, this will result in[๐]๐ฃโฒ being diagonal or as close to being diagonal as possible (block diagonal or Jordan form).
Let us apply this to the second example in the previous section. In that example, we had
[๐]๐ฃ =
โกโขโขโขโขโขโขโฃ
1
โ2โ 1
โ21
โ21
โ2
โคโฅโฅโฅโฅโฅโฅโฆ
The eigenvectors of [๐]๐ฃ areโกโขโขโขโขโฃโ๐1
Which is diagonal representation. Hence we see that the linear transformation ๐ is represented asโกโขโขโขโขโขโขโขโฃ
๏ฟฝ 12 โ
12 ๐๏ฟฝโ2 0
0 ๏ฟฝ 12 +
12 ๐๏ฟฝโ2
โคโฅโฅโฅโฅโฅโฅโฅโฆw.r.t. basis ๐ฃโฒ
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1.3 Summary of similarity transformation
To obtain ๐ต = ๐โ1๐ด๐ such that ๐ต is real and diagonal requires that ๐ด be real and symmetric. Theeigenvalues of ๐ด goes into the diagonal of ๐ต and the eigenvectors of ๐ด go into the columns of ๐.This is an algebraic view.
Geometrically,๐ด is viewed as the matrix representation under some basis ๐ฃ of a linear transformation๐. And ๐ต is the matrix representation of the same linear transformation ๐ but now under a newbasis ๐ฃโฒ and ๐ is the matrix that represents the change of basis from ๐ฃโฒ to ๐ฃ.
The question then immediately arise: If ๐ดmust be real and symmetric (for ๐ต to be real and diagonal),what does this mean in terms of the linear transformation ๐ and change of basis matrix ๐? Thisclearly mean that, under some basis ๐ฃ, not every linear transformation represented by ๐ด can have asimilar matrix ๐ต which is real and diagonal. Only those linear transformations which result in realand symmetric representation can have a similar matrix which is real and diagonal. This is shownin the previous examples, where we saw that ๐ defined as rotation by 450 under the standard basis
๐ฃ =
โงโชโชโจโชโชโฉ
โกโขโขโขโขโฃ10
โคโฅโฅโฅโฅโฆ ,โกโขโขโขโขโฃ01
โคโฅโฅโฅโฅโฆ
โซโชโชโฌโชโชโญresulted in ๐ด =
โกโขโขโขโขโขโขโฃ
1
โ2โ 1
โ21
โ21
โ2
โคโฅโฅโฅโฅโฅโฅโฆ and since this is not symmetric, hence we will not be able
to factor this matrix using similarity transformation to a diagonal matrix, no matter which changeof basis we try to represent ๐ under. The question then, under a given basis ๐ฃ what is the class oflinear transformation which leads to a matrix representation that is symmetric? One such examplewas shown above which is the projection into the line at 1350. This question needs more time tolook into.
We now write ฮ to represent a diagonal matrix, hence the similarity transformation above can bewritten as
ฮ = ๐โ1๐ด๐
Or๐ด = ๐ฮ๐โ1
A M M1
Eigenvalues of A go into the diagonal
Eigenvectors of A go into the columns
A is real and symmetric
2 Singular value decomposition (SVD)
Using similarity transformation, we found that for a real and symmetric matrix ๐ด we are able todecompose it as ๐ด = ๐ฮ๐โ1 where ฮ is diagonal and contains the eigenvalues of ๐ด on the diagonal,and ๐ contains the right eigenvectors of ๐ด in its columns.
2.1 What is right and left eigenvectors?
Right eigenvectors are the standard eigenvectors we have been talking about all the time. When ๐is an eigenvalues of ๐ด then ๐ is a right eigenvector when we write
๐ด๐ = ๐๐
However, ๐ is a left eigenvector of ๐ด when we have
๐๐ป๐ด = ๐๐๐ป
where ๐๐ป is the Hermitian of ๐
2.2 SVD
Given any arbitrary matrix ๐ด๐ร๐ it can be factored into 3 matrices as follows ๐ด๐ร๐ = ๐๐ร๐๐ท๐ร๐๐๐ร๐where ๐ is a unitary matrix (๐๐ป = ๐โ1 or ๐๐ป๐ = ๐ผ), and ๐ is also unitary matrix.
These are the steps to do SVD
1. Find the rank of ๐ด, say ๐
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2. Let ๐ต = ๐ด๐ป๐ร๐๐ด๐ร๐, hence ๐ต๐ร๐ is a square matrix, and it is semi positive definite, hence its
eigenvalues will all be โฅ 0. Find the eigenvalues of ๐ต, call these ๐2๐ . There will be ๐ sucheigenvalues since ๐ต is of order ๐ร๐. But only ๐ of these will be positive, and ๐โ ๐ will be zero.Arrange these eigenvalues such that the first ๐ non-zero eigenvalues come first, followed by
3. Initialize matrix ๐ท๐ร๐ to be all zeros. Take the the first ๐ eigenvalues from above (non-zero
ones), and take the square root of each, hence we get๐ singular values
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐1, ๐2,โฏ , ๐๐ ,and write these down thediagonal of ๐ท starting at ๐ท (1, 1), i.e. ๐ท (1, 1) = ๐1, ๐ท (2, 2) = ๐2,โฏ ,๐ท (๐, ๐) = ๐๐. Notice that thematrix ๐ท need not square matrix. Hence we can obtain an arrangement such as the followingfor ๐ = 2
๐ท =
โโโโโโโโโโโ
๐1 0 0 00 ๐2 0 00 0 0 0
โโโโโโโโโโโ
where the matrix ๐ด was 3 ร 4 for example.
4. Now find each eigenvalue of ๐ด๐ป๐ด. For each eigenvalue, find the corresponding eigenvector.Call these eigenvectors ๐1, ๐2,โฏ , ๐๐.
5. Normalize the eigenvectors found in the above step. Now ๐1, ๐2,โฏ , ๐๐ eigenvector will bean orthonormal set of vectors. Take the Hermitian of each eigenvector ๐๐ป1 , ๐๐ป2 ,โฏ , ๐๐ป๐ andmake one of these vectors (now in row format instead of column format) go into a rowin the matrix ๐.i.e. the first row of ๐ will be ๐๐ป1 , the second row of ๐ will be ๐๐ป2 , etc...
6. Now we need to find a set of ๐ orthonormal vectors, these will be the columns of the matrix๐: find a set of ๐ orthonormal eigenvector ๐๐ =
1๐๐๐ด๐๐, for ๐ = 1โฏ๐. Notice that here we only
use the first ๐ vectors found in step 5. Take each one of these ๐๐ vectors and make them intocolumns of ๐. But since we need ๐ columns in ๐ not just ๐, we need to come up with ๐ โ ๐more basis vectors such that all the ๐ vectors form an orthonormal set of basis vectors forthe row space of ๐ด, i.e. โ๐. If doing this by hand, it is easy to find the this ๐โ ๐ by inspection.In a program, we could use the same process we used with Gram-Schmidt, where we learnedhow find a new vector which is orthonormal to a an existing set of other vectors. Anotherway to find the matrix ๐ is to construct the matrix ๐ด๐ด๐ป and find its eigenvectors, those gointo the columns of the matrix ๐ as follows