λ³-(a11+a22+a33)λ² +(a11*a22+a22*a33+a33*a11-a23*a32-a12*a21-a13*a31)λ+[(a11*a23*a32+a22*a13*a31+a33*a12*a21)-(a12*a23*a31+a21*a13*a32) - a11*a22*a33] =0

or

λ³-(Σa_{ii i=1,2,3})λ²+ (Σa_iia_{jj  i≠j, i,j=1,2,3} - Σa_ija_{ji  i≠j, i,j=1,2,3} )λ + [(Σa_iia_jka_{kj i≠j≠k i,j,k=1,2,3})-( Σa_ija_jka_{ki   +} Σa_jia_ika_{kj ; i≠j≠k} i,j,k=1,2,3 )-(∏a_{ii  i=1,2,3})] = 0

or 

λ³ -     TrA*λ² +              ( λ1 λ2+ λ2 λ3+ λ3 λ1) λ                 +  λ1 λ2 λ3(determinant) =0

or

 λ³+bλ²  +cλ +d=0

Δ₀=b²-3c=(trace)²-3( Σa_iia_{jj  i≠j, i,j=1,2,3} - Σa_ija_{ji  i≠j, i,j=1,2,3} )

Δ₁=tr*3²(Σa_iia_{jj  i≠j, i,j=1,2,3} - Σa_ija_{ji  i≠j, i,j=1,2,3} )-2(trace)³- 3³*D

Orthogonality of 3x3 Matrix:

Let the matrix be A=

a11 a12 a13

a21 a22 a23

a31 a32 a33

Then A^T is

a11 a21 a31

a12 a22 a32

a13 a23 a33

If A is orthogonal , then AA^T =K =I which means

a11² +a12²  +a13²                                            a11*a21+a12*a22+a13*a23                                      a11*a31+a12*a32+a13*a33

a21*a11+a22*a12+a23*a13                            a21² +a22²  +a23²                                                      a21*a31+a22*a32+a23*a33

a31*a11+a32*a12+a33*a13                            a31*a21+a32*a22+a33*a23                                      a31² +a32²  +a33²

(1a) The product matrix K is a symmetric matrix. K=

K11        K12    K13

K12        K22    K23

K13        K23    K33

(any square matrix multiplied/ added  by its transpose becomes a symmetric matrix )

and

 (2a)   K11=a11² +a12²  +a13²   =  1

          K22=a21² +a22²  +a23²   =  1

          K33=a31² +a32²  +a33²   =  1

          K12=K21=a11*a21+a12*a22+a13*a23=0

          K13=K31=a11*a31+a12*a32+a13*a33=0

          K23=K32=a21*a31+a22*a32+a23*a33=0

A^TA =tK =I which means

a11² +a21²  +a31²                                            a11*a12+a21*a22+a31*a32                                      a11*a13+a21*a23+a31*a33

a11*a12+a21*a22+a31*a32                            a12² +a22²  +a32²                                                      a12*a13+a22*a23+a32*a33

a11*a13+a21*a23+a31*a33                            a12*a13+a22*a23+a32*a33                                      a13² +a23²  +a33²

(1a) The product matrix tK is a symmetric matrix. K=

tK11        tK12    tK13

tK12        tK22    tK23

tK13        tK23    tK33

(any square matrix multiplied/ added  by its transpose becomes a symmetric matrix )

and

 (2a)   tK11=a11² +a21²  +a31²   =  1

          tK22=a12² +a22²  +a32²   =  1

          tK33=a13² +a23²  +a33²   =  1

          tK12=tK21=a11*a12+a21*a22+a31*a32=0

          tK13=tK31=a11*a13+a21*a23+a31*a33=0

          tK23=tK32=a12*a13+a22*a23+a32*a33=0

Comparing  orthogonal matrices

         2x2 matrices                                      3x3 matrices                            

a11² +a12²     =  1                          a11² +a12²  +a13²   =  1                          The blue portion is a consequence of AA^T =K =I

a21² +a22²    =  1                           a21² +a22²  +a23²   =  1

---------------------                            a31² +a32²  +a33²   =  1

a11² +a21²    =  1                           a11² +a21²  +a31²   =  1                          The red portion is a consequence of A^TA =tK =I

a12² +a22²   =  1                            a12² +a22²  +a32²   =  1

--------------------                             a13² +a23²  +a33²   =  1

a11*a21+a12*a22=0                     a11*a21+a12*a22+a13*a23=0                 The blue portion is a consequence of AA^T =K =I

-------------------------                     a11*a31+a12*a32+a13*a33=0

-------------------------                     a21*a31+a22*a32+a23*a33=0

a11*a12+a21*a22=0                    a11*a12+a21*a22+a31*a32=0                    The red portion is a consequence of A^TA =tK =I

-------------------------                     a11*a13+a21*a23+a31*a33=0

-------------------------                     a12*a13+a22*a23+a32*a33=0

(3a) dimension is 1                         In general, dimension is 3 since there are three variable angles corresponding to 3 rotations . It can be 2 or 1 with 2

                                                      rotation angles or even 1 rotation angle.

(4a) vector norm preserved.           vector norm preserved

(5a) matrices commute                   In general, matrices do not commute.

(6a) eigen value both complex   eigen value 3 real,or 1 real & 2 complex.

 for rotation, real for reflection.

(7a) determinant +1 or -1           determinant +1 or -1

(8a) Signature of signs(odd-even)  Signature of signs +,- is (7,2),(2,7),(6,3) , (3,6) or (5,4),(4,5) & in at least 2 rows as well as in 2 columns, signature

                                                       to be++-(1,2) ,--+ (2,1)....

        i.e +++-(3,1) or ---+(1,3)       If there are zeros, half of them to be treated as +, other half - so that the configuration becomes (6,3) or (3,6).

     if 0 are there, half of them to     With odd no. of zeros, the odd one  to be treated as +, or - so that the configuration becomes (6,3) or (3,6).

       be treated +,half -                    This postulate to be verified to find out whether true or not 

(9a) product of orthogonal             product of orthogonal  matrices is orthogonal

       matrices is orthogonal  

(10a) |a_ij | = |a_ji  |                           orthogonal matrices are not necessarily symmetric in all cases nor in general  |a_ij | = |a_ji  | .

(3) We have found that the 3 eigen values of a 3x3 matrix are either all real or one is real & the rest two are complex conjugates. Thus at least 1 eigen value is real. For orthogonal matrix, at least 1 of the real eigen values is +1 or -1 since determinant is the product of eigen values and in orthogonal case, determinant is +1 or -1.(we have , however, not yet proved that determinant of orthogonal matrix is ±1)

(4) Logically assuming ( because we have not yet proved)  that determinant of orthogonal matrix is ±1, the eigen value combination can be any of the below :

(1,1,1), (-1,-1,-1), (1,1,-1),(-1,1,1) ...... if all are real

(ω,ω²),(-ω,-ω²),(ω,-ω²),(-ω,ω²)....if 2 are complex conjugates where ω , ω²   are cube roots of unity. The real part can be +1 or -1.

ω = (-1 +√3i) / 2 = -0.5 +0.8660i

ω² =(-1 -√3i) / 2 =-0.5  - 0.8660i 

Cases where 3x3 matrix has one dimensional Representation:

Rotation about z-axis : Rz=

a11  a12  0

a21  a22  0

0       0     1

Rotation about y-axis : Ry=

a11  0   a13

0      1    0

a31  0   a33

Rotation about x-axis : Rx=

1      0     0

0    a22  a23

0    a32  a33

In 1-dimensional representation, trace=±1±2cosθ  or ±1±2sinθ. Hence maximum value is +3 and minimum value is -3.

Exa-

-1 0 0       orthogonal, trace=-3

0 -1 0

0  0 -1

1 0 0              orthogonal,  trace=3

0 1 0

0 0 1

0 -1  0            orthogonal, trace=-1

1  0  0

0  0 -1

(1/3)   1 -2  2  orthogonal, trace=0.333

           2-1 -2

           2 2  1

* In terms of modern mathematics, rotations are distance and orientation preserving transformations in 3-dimensional Euclidean (affine) space which have a fixed point. Such transformations are associated with linear operators on the difference space R³ that preserve inner product (are isometric) and preserve orientation (have unit determinant). In an orthogonal basis of these operators correspond one-to-one with orthogonal 3 × 3 matrices with determinant +1. Since for such (non-identity) matrices exactly one eigenvector has eigenvalue +1, this eigenvector gives the direction of the axis. The product of two orthogonal matrices is again orthogonal, and from the determinant rule: det(AB) = det(A)det(B) follows that the product matrix has also unit determinant. The matrix product being associative and the inverse of an orthogonal matrix being orthogonal, the matrices form a group of infinite order, commonly denoted by SO(3), the special (det = 1) orthogonal  group in 3 dimensions. Note that the map A (matrix) → det(A) {this is a number} is a group homomorphism: the set of determinants forms a 1 dimensional irreducible representation (the identity representation) of SO(3). 

A rotation matrix R has at least one invariant vector n, i.e., R n = n. Note that this is equivalent to stating that the vector n is an eigenvector of the matrix R with eigen value λ = 1. A proper rotation matrix R has at least one unit eigenvalue. Using the two relations: we find From this follows that λ = 1 is a root (solution) of the secular equation, that is, In other words, the matrix R − E (identity matrix) is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say n, for which R.n=n

*An m×m matrix A has m orthogonal eigenvectors if and only if A is normal, that is, if A†A = AA†.

* Rotations in Euler Angles & Rotations in Fixed Angles :

In Euler angles, the each rotation is imagined to be represented in the post-rotation coordinate frame of the last rotation

In Fixed angles, all rotations are imagined to be represented in the original (fixed) coordinate frame.

ZYX Euler angles can be thought of as:

1. ZYX Euler

2. XYZ Fixed

R_zyx =R_z (φ)R_y (θ)R_x (ψ) ..... in Euler Angle Rotation ( right multiplication of subsequent rotations, applicable to rotating frame)

R_zyx =R_x (ψ)R_y (θ)R_z (φ)..... in Fixed Angle Rotations ( left multiplication of subsequent rotations, applicable to fixed frame)

* Links: 1,

Preservation of Inner Product under Proper Rotation (Isometry):

Suppose A is a vector and B is another vector such that A.B=k (a scalar quantity) . Now there is a linear transformation of A by an operator X with matrix elements represented by Xij which is a proper rotation matrix. 

 XA=A', XB=B' . It will be seen that A'.B' = k

Preservation of Orientation under Proper Rotation :

Now, since the determinant is 1, there will be at least 1 eigen value having value +1 or -1. Corresponding to this , there will be an eigen vector which shall remain invariant under rotation. This eigen vector represents the axis of rotation and direction of axis.

Matrices representing Rotation as well as translation :

Any displacement in space can be broken into a rotation and a translation along the line joining the origin and the new rotated point. If the corresponding operator has to be found out which is represented as a square matrix, that is a bit tricky. For example, in 2-D Euclidian Space, if a rotation matrix is constructed , say M=a   -b   , translation cannot be accommodated there. To do so, we have to construct a 3x3 matrix like

                                  b    a

M'=

a  -b    0

b   a    0

0   0    1

which forms an inhomogeneous equation.

if the vector in 2-D is X= x , then MX = X'   where X'= x'  and x'=ax-by & y'=bx+ay

                                         y                                              y'

but with translation built in, it becomes M'X=X' where X= x    and X' =x'

                                                                                              y                  y'

                                                                                              1                  1

1 remaining unchanged upon transformation which indicates there is no translation. If there is translation , in matrix M', instead of 1, there will be a scalar other than 1. The representation of X in this case is X=xi+yj+1 which is a trinion representation which in a 3-D case, becomes quaternion . This is a convenient representation of a vector to accommodate translation as well as rotation.

Leading Principal Sub Matrix of Order k of a nxn Matrix :

Leading Principal Sub-Matrix of order k is obtained by deleting last n-k rows and columns of the matrix

Example :A

2   -1   0

-1  2  -1

0  -1   2

Leading Principal Sub-Matrix of order 1: delete last 3-1=2 rows and columns

|A1| =2                            

Leading Principal Sub-Matrix of order 2: delete last 3-2=1 row and column

A2= 2  -1   |A2| = 3

       -1   2

Leading Principal Sub-Matrix of order 3: delete last 3-3=0 rows and columns

A3 =

2   -1   0   |A3| =6+(-2) =4

-1  2  -1

0  -1   2

Spectrum of a Matrix:

It is the set of all eigen values

Pseudo Determinant of a Singular Matrix:

is the product of all non-zero eigen values.

Positive Definite Matrix A:

A is positive definite if x^TA x > 0 for all x ≠ 0 where x is non-zero vector of n dimension and A is a square matrix of order n. All its eigen values are positive.

Pivots of a Matrix:

Pivots are the first non-zero element in each row in the elimination matrix in Row Echelon Form / RREF . Rank of a matrix is the number of pivots in its reduced elimination form. Rank of a matrix also is the number of linearly independent row/column vectors in the matrix.

Row Echelon Form :

(1) All non zero rows are above all zero rows.

(2) Each leading entry of a row is in a column to the right of the leading entry of the row above it. In other words, row leading entries are in stair case form.

Example:

1  2  3   5

0  0  1   2

0 0   0  4

0  0  0  0

Representation of the triangle in 3x3 matrix:

Let a,b,c be the three sides of a triangle and s be its semi-perimeter i.e. s =  (a+b+c)/2

then area A of the triangle : A²=  s(s-a)(s-b)(s-c)

Now we design a diagonal matrix  D=

s-a   0    0

0    s-b   0

0     0    s-c

its determinant Δ = (s-a)(s-b)(s-c)

its trace,          tr  = s

and      A² = tr * Δ

  perimeter= 2 * tr

Eigen Values are

λ1 = s-a  or    s-a = λ1

λ2 = s-b  or    s-b = λ2

λ3 = s-c  or    s-c = λ3

Side a =tr - λ1

Side b =tr - λ2

Side c =tr - λ3

For being eligible to be the sides of the triangle,

Case 1 :

all 3 eigen values have to be positive ...... matrix has to be positive definite. trace to be +ve and Δ =+ve

or  

all 3 eigen values have to be negative ...... matrix has to be negative definite. trace to be -ve and Δ =-ve

Trace and Determinant to be of same sign.

Case 2 :

Any 2 eigen values -Ve, 1 eigen value +Ve , such that modulus of sum of 2 negative eigen values > modulus of 1 positive eigen value

or

Any 2 eigen values +Ve, 1 eigen value -Ve , such that modulus of  negative eigen value > modulus of sum of 2 positive eigen values

Hence any 3x3 non singular square positive definite or negative definite matrix can represent a triangle.

Case 2 is very interesting and to be explored sepatrately.

let the matrix A=

1    1     1

1   ω²   ω

1   ω    ω²

det=3ω(1-ω) ; trace=1+ω

After elementary row operation R1 <->R1+R2+R3

A'=

3    0    0

1   ω²   ω

1   ω    ω²

det=3ω(1-ω) ; trace=3+2ω²  =1-2ω

and if transformation matrix is T, then TA = A' . T is

1  1   1

0  1   0

0  0   1

det= 1 ; trace =3 and it is an upper triangular matrix.

if AT' = A', then T'=

5/3   -1/3    -1/3

2/3    2/3    -1/3

2/3   -1/3     2/3

det=1;  trace=3