Ovde rešavamo zadatke iz MML Book

MML-book Chapter 4. Matrix Decompositions, Exercises

Koristićemo R i matlib biblioteku.

library(matlib)

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4.1

\(\begin{vmatrix} 1 & 3 & 5\\ 2 & 4 & 6\\ 0 & 2 & 4\\ \end{vmatrix} = 1 \cdot \begin{vmatrix} 4 & 6\\ 2 & 4\\ \end{vmatrix} -3 \cdot \begin{vmatrix} 2 & 6\\ 0 & 4\\ \end{vmatrix} + 5 \cdot \begin{vmatrix} 2 & 4\\ 0 & 2\\ \end{vmatrix} =\) \(4 \cdot 4 - 2 \cdot 6 - 3 \cdot ( 2 \cdot 4 -0 \cdot 6 ) + 5 \cdot ( 2 \cdot 2 - 0 \cdot 4 ) = 0\)

#########
#  4.1  #
#########

A = matrix(c(1,3,5,2,4,6,0,2,4),ncol = 3, byrow= "T"); det(A)
## [1] 0

4.2

Razvićemo po drugoj koloni: \(\begin{vmatrix} 2 & 0 & 1 & 2 & 0\\ 2 & -1 & 0 & 1 & 1\\ 0 & 1 & 2 & 1 & 2\\ -2 & 0 & 2 & -1 & 2\\ 2 & 0 & 0 & 1 & 1\\ \end{vmatrix} = -1 \cdot \begin{vmatrix} 2 & 1 & 2 & 0\\ 0 & 2 & 1 & 2\\ -2 & 2 & -1 & 2\\ 2 & 0 & 1 & 1\\ \end{vmatrix} - 1 \cdot \begin{vmatrix} 2 & 1 & 2 & 0\\ 2 & 0 & 1 & 1\\ -2 & 2 & -1 & 2\\ 2 & 0 & 1 & 1\\ \end{vmatrix} = \cdots = 6\)

#########
#  4.2  #
#########

A = matrix(c(2,0,1,2,0,2,-1,0,1,1,0,1,2,1,2,-2,0,2,-1,2,2,0,0,1,1),ncol = 5);  det(A)
## [1] 6

4.3

  1. \(\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \lambda \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \Leftrightarrow \begin{bmatrix} 1-\lambda & 0 \\ 1 & 1-\lambda \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow \begin{vmatrix} 1-\lambda & 0 \\ 1 & 1-\lambda \end{vmatrix} = (1 - \lambda )^2 = 0 \Leftrightarrow \lambda_{1,2} = 1\) i \(\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow x_1 = 0\) što daje jednodimenzionalni prostor koji odgovara \(\lambda = 1: E_{1} = \mbox{span} ( [0, 1]^T )\).
  2. \(\begin{bmatrix} -2 & 2 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \lambda \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \Leftrightarrow \begin{bmatrix} -2-\lambda & 2 \\ 2 & 1-\lambda \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow \begin{vmatrix} -2-\lambda & 2 \\ 2 & 1-\lambda \end{vmatrix} = \lambda^2 + \lambda - 6 = 0\)
    Dobijamo \(\lambda = 2 \vee \lambda = -3\).
    Za \(\lambda = 2\): \(\begin{bmatrix} -4 & 2 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow x_2 = 2 x_1\) što daje jednodimenzionalni prostor \(E_{2} = \mbox{span} ( [1, 2]^T )\).
    Za \(\lambda = -3\): \(\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow x_1 = - 2 x_2\) što daje jednodimenzionalni prostor \(E_{-3} = \mbox{span} ( [-2, 1]^T )\).
#########
#  4.3  #
#########

# b
A = matrix(c(-2,2,2,1),ncol = 2, byrow = "T"); A
##      [,1] [,2]
## [1,]   -2    2
## [2,]    2    1
eigen(A)->ea; ea
## eigen() decomposition
## $values
## [1]  2 -3
## 
## $vectors
##            [,1]       [,2]
## [1,] -0.4472136 -0.8944272
## [2,] -0.8944272  0.4472136
round(ea$vectors %*% diag(-c(sqrt(5),-sqrt(5))),digits=8)
##      [,1] [,2]
## [1,]    1   -2
## [2,]    2    1

4.4

#########
#  4.4  #
#########

A = matrix(c(0,-1,1,1,-1,1,-2,3,2,-1,0,0,1,-1,1,0),ncol = 4, byrow = "T"); A
##      [,1] [,2] [,3] [,4]
## [1,]    0   -1    1    1
## [2,]   -1    1   -2    3
## [3,]    2   -1    0    0
## [4,]    1   -1    1    0
eigen(A)->ea; ea
## eigen() decomposition
## $values
## [1]  2+0.000000e+00i  1+0.000000e+00i -1+2.720552e-08i -1-2.720552e-08i
## 
## $vectors
##                 [,1]   [,2]                        [,3]
## [1,] 5.773503e-01+0i 0.5+0i -3.042967e-16-1.923721e-08i
## [2,] 2.319703e-16+0i 0.5+0i -7.071068e-01+0.000000e+00i
## [3,] 5.773503e-01+0i 0.5+0i -7.071068e-01+1.923721e-08i
## [4,] 5.773503e-01+0i 0.5+0i  8.012345e-16+8.740850e-24i
##                             [,4]
## [1,] -3.042967e-16+1.923721e-08i
## [2,] -7.071068e-01+0.000000e+00i
## [3,] -7.071068e-01-1.923721e-08i
## [4,]  8.012345e-16-8.740850e-24i
zapsmall(ea$values)
## [1]  2+0i  1+0i -1+0i -1+0i
lambda=Re(ea$values); zapsmall(lambda)
## [1]  2  1 -1 -1
vecs=Re(ea$vectors); zapsmall(vecs)
##           [,1] [,2]       [,3]       [,4]
## [1,] 0.5773503  0.5  0.0000000  0.0000000
## [2,] 0.0000000  0.5 -0.7071068 -0.7071068
## [3,] 0.5773503  0.5 -0.7071068 -0.7071068
## [4,] 0.5773503  0.5  0.0000000  0.0000000
vecs=vecs %*% diag(c(sqrt(3),2,-sqrt(2),-sqrt(2)))
round(vecs,digits=8)
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    0    0
## [2,]    0    1    1    1
## [3,]    1    1    1    1
## [4,]    1    1    0    0

Vidimo da su karakteristične vrednosti i karakteristični prostori koji im odgovaraju:
\(\lambda = 2\), \(E_{2} = \mbox{span} ( [1,0,1,1]^T )\),
\(\lambda = 1\), \(E_{1} = \mbox{span} ( [1,1,1,1]^T )\),
\(\lambda = -1\) algebarski dvostruki, geometrijski jednostruki: \(E_{-1} = \mbox{span} ( [0,1,1,0]^T )\).

4.5

  • \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \mbox{diag} ( 1, 1 )\) invertibilna i dijagonalizabilna

  • \(\mbox{det} ( \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} ) = 0\), \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \mbox{diag} ( 1, 0 )\), nije invertibilna, jeste dijagonalizabilna

  • \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}^{-1} = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}\)
    \(\mbox{det} ( \begin{bmatrix} 1-\lambda & 1 \\ 0 & 1-\lambda \end{bmatrix} ) = 0 \Rightarrow \lambda = 1\) dvostruka nula.
    Za \(\lambda = 1\): \(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow x_2 = 0\) što daje jednodimenzionalni prostor \(E_{1} = \mbox{span} ( [1, 0]^T )\).
    Jeste invertibilna, nije dijagonalizabilna

  • \(\mbox{det} ( \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} ) = 0\),
    \(\mbox{det} ( \begin{bmatrix} -\lambda & 1 \\ 0 & -\lambda \end{bmatrix} ) = 0 \Rightarrow \lambda = 0\) dvostruka nula.
    Za \(\lambda = 0\): \(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \Leftrightarrow x_2 = 0\) što daje jednodimenzionalni prostor \(E_{0} = \mbox{span} ( [1, 0]^T )\).
    Nije invertibilna, nije dijagonalizabilna

4.6

#########
#  4.6  #
#########

# a

A = matrix(c(2,1,0,3,4,0,0,3,1), ncol = 3); A
##      [,1] [,2] [,3]
## [1,]    2    3    0
## [2,]    1    4    3
## [3,]    0    0    1
ev = eigen(A)
ev$values
## [1] 5 1 1
ev$vectors
##            [,1]       [,2]          [,3]
## [1,] -0.7071068 -0.9486833  9.486833e-01
## [2,] -0.7071068  0.3162278 -3.162278e-01
## [3,]  0.0000000  0.0000000  9.362223e-17
zapsmall(ev$vectors)
##            [,1]       [,2]       [,3]
## [1,] -0.7071068 -0.9486833  0.9486833
## [2,] -0.7071068  0.3162278 -0.3162278
## [3,]  0.0000000  0.0000000  0.0000000
round(matrix(ev$vectors,ncol=3) %*% diag(-c(sqrt(2),sqrt(10),-sqrt(10))),digits=8)
##      [,1] [,2] [,3]
## [1,]    1    3    3
## [2,]    1   -1   -1
## [3,]    0    0    0
  1. Matrica \(A\) nije dijagonazibilna. Sopstveni prostori koji odgovaraju karakterističnim vrednostima su:
    Za \(\lambda = 5\): \(E_5 = \mbox{span} ( [1, 1, 0 ]^T)\) i
    Za \(\lambda = 1\) dvostruki algebarski: jednostruki geometrijski \(E_1 = \mbox{span} ( [3, -1, 0 ]^T)\)
#########
#  4.6  #
#########

# b

A = matrix(c(1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0), ncol = 4, byrow = T ); A
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    0    0
## [2,]    0    0    0    0
## [3,]    0    0    0    0
## [4,]    0    0    0    0
ev=eigen(A)
round(ev$values, digits = 8)
## [1] 1 0 0 0
round(ev$vectors, digits = 8)
##      [,1]       [,2] [,3] [,4]
## [1,]    1 -0.7071068    0    0
## [2,]    0  0.7071068    0    0
## [3,]    0  0.0000000    1    0
## [4,]    0  0.0000000    0    1
evx = round(ev$vectors %*% diag(c(1,-sqrt(2),1,1)), digits = 8); evx
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    0    0
## [2,]    0   -1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1
evx1 = solve(evx); round(evx1, digits = 8 )
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    0    0
## [2,]    0   -1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1
evx %*% evx1
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1
evx1 %*% diag(c(1,0,0,0)) %*% evx
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    0    0
## [2,]    0    0    0    0
## [3,]    0    0    0    0
## [4,]    0    0    0    0
  1. Matrica \(A\) je dijagonazibilna. Sopstveni prostori koji odgovaraju karakterističnim vrednostima su:
    Za \(\lambda = 1\): \(E_5 = \mbox{span} ( [1, 0, 0, 0 ]^T)\) i
    Za \(\lambda = 0\) trostruki algebarski i trostruki geometrijski: \(E_0 = \mbox{span} ( [1, -1, 0, 0 ]^T, [ 0, 0, 1, 0 ]^T, [ 0, 0, 0, 1 ]^T )\)

4.7

#########
#  4.7  #
#########

# a

A = matrix(c(0,-8,1,4), ncol = 2); A
##      [,1] [,2]
## [1,]    0    1
## [2,]   -8    4
ev=eigen(A);
round(ev$values, digits = 8)
## [1] 2+2i 2-2i
  1. Matrica \(A\) nije dijagolizabilna (nad \(\mathbb{R}\)) jer nema korene nad skupom \(\mathbb{R}\).
#########
#  4.7  #
#########

# b

A = matrix(c(1,1,1,1,1,1,1,1,1), ncol = 3); A
##      [,1] [,2] [,3]
## [1,]    1    1    1
## [2,]    1    1    1
## [3,]    1    1    1
ev=eigen(A)
round(ev$values, digits = 8)
## [1] 3 0 0
round(ev$vectors, digits = 8)
##            [,1]       [,2]       [,3]
## [1,] -0.5773503  0.8164966  0.0000000
## [2,] -0.5773503 -0.4082483 -0.7071068
## [3,] -0.5773503 -0.4082483  0.7071068
p=round(ev$vectors %*% diag(c(-sqrt(3),sqrt(6),sqrt(2))), digits = 8); p
##      [,1] [,2] [,3]
## [1,]    1    2    0
## [2,]    1   -1   -1
## [3,]    1   -1    1
p1=solve(p); p1
##           [,1]       [,2]       [,3]
## [1,] 0.3333333  0.3333333  0.3333333
## [2,] 0.3333333 -0.1666667 -0.1666667
## [3,] 0.0000000 -0.5000000  0.5000000
zapsmall(6*p1)
##      [,1] [,2] [,3]
## [1,]    2    2    2
## [2,]    2   -1   -1
## [3,]    0   -3    3
p %*% diag(c(3,0,0))%*% p1
##      [,1] [,2] [,3]
## [1,]    1    1    1
## [2,]    1    1    1
## [3,]    1    1    1
  1. Matrica \(A\) je dijagolizabilna:
    \(\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix} = \begin{bmatrix} 1 & 2 & 0 \\ 1 & -1 & -1 \\ 1 & -1 & 1 \\ \end{bmatrix} \begin{bmatrix} 3 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \frac16 \begin{bmatrix} 2 & 2 & 2 \\ 2 & -1 & -1 \\ 0 & -3 & 3 \\ \end{bmatrix}\),
    gde je \(\frac16 \begin{bmatrix} 2 & 2 & 2 \\ 2 & -1 & -1 \\ 0 & -3 & 3 \\ \end{bmatrix} = \begin{bmatrix} 1 & 2 & 0 \\ 1 & -1 & -1 \\ 1 & -1 & 1 \\ \end{bmatrix}^{-1}\).
#########
#  4.7  #
#########

# c

A = matrix(c(5,0,-1,1,4,1,-1,1,2,-1,3,-1,1,-1,0,2), ncol = 4); A
##      [,1] [,2] [,3] [,4]
## [1,]    5    4    2    1
## [2,]    0    1   -1   -1
## [3,]   -1   -1    3    0
## [4,]    1    1   -1    2
ev = eigen(A)
ev$values
## [1] 4 4 2 1
ev$vectors
##            [,1]       [,2]          [,3]          [,4]
## [1,] -0.5773503  0.5773503 -5.773503e-01 -7.071068e-01
## [2,]  0.0000000  0.0000000  5.773503e-01  7.071068e-01
## [3,]  0.5773503 -0.5773503  4.079220e-16  1.480385e-16
## [4,] -0.5773503  0.5773503 -5.773503e-01  4.261214e-16
zapsmall(ev$vectors)
##            [,1]       [,2]       [,3]       [,4]
## [1,] -0.5773503  0.5773503 -0.5773503 -0.7071068
## [2,]  0.0000000  0.0000000  0.5773503  0.7071068
## [3,]  0.5773503 -0.5773503  0.0000000  0.0000000
## [4,] -0.5773503  0.5773503 -0.5773503  0.0000000
round(matrix(ev$vectors,ncol=4) %*% diag(c(-sqrt(3),sqrt(3),-sqrt(3),-sqrt(2))),digits=8)
##      [,1] [,2] [,3] [,4]
## [1,]    1    1    1    1
## [2,]    0    0   -1   -1
## [3,]   -1   -1    0    0
## [4,]    1    1    1    0
  1. Matrica \(A\) nije dijagonazibilna. Sopstveni prostori koji odgovaraju karakterističnim vrednostima su:
    Za \(\lambda = 4\) dvostruki algebarski: jednostruki geometrijski \(E_4 = \mbox{span} ( [1, 0, -1, 1 ]^T)\),
    za \(\lambda = 2\): \(E_2 = \mbox{span} ( [ 1, 1, -1, 0, 1 ]^T)\) i
    za \(\lambda = 1\): \(E_1 = \mbox{span} ( [ 1, -1, 0, 0 ]^T)\).
#########
#  4.7  #
#########

# d

A = matrix(c(5,-1,3,-6,4,-6,-6,2,-4), ncol = 3); A
##      [,1] [,2] [,3]
## [1,]    5   -6   -6
## [2,]   -1    4    2
## [3,]    3   -6   -4
ev=eigen(A)
round(ev$values, digits = 8)
## [1] 2 2 1
round(ev$vectors, digits = 8)
##            [,1]       [,2]       [,3]
## [1,] -0.6854346 -0.9414664 -0.6882472
## [2,]  0.3141575 -0.1976432  0.2294157
## [3,] -0.6568748 -0.2730900 -0.6882472
p=round(ev$vectors, digits = 8); p
##            [,1]       [,2]       [,3]
## [1,] -0.6854346 -0.9414664 -0.6882472
## [2,]  0.3141575 -0.1976432  0.2294157
## [3,] -0.6568748 -0.2730900 -0.6882472
p1=solve(p); p1
##           [,1]       [,2]      [,3]
## [1,] -4.016414  9.2993609  7.116201
## [2,] -1.324541 -0.3973624  1.192087
## [3,]  4.358899 -8.7177980 -8.717798
round(p %*% diag(c(2,2,1))%*% p1,digits=8)
##      [,1] [,2] [,3]
## [1,]    5   -6   -6
## [2,]   -1    4    2
## [3,]    3   -6   -4
  1. Matrica \(A\) je dijagolizabilna.

4.8

#########
#  4.8  #
#########

A = matrix(c(3,2,2,3,2,-2), ncol = 3); A
##      [,1] [,2] [,3]
## [1,]    3    2    2
## [2,]    2    3   -2
svda = svd(A, nu=2, nv = 3); svda
## $d
## [1] 5 3
## 
## $u
##            [,1]       [,2]
## [1,] -0.7071068 -0.7071068
## [2,] -0.7071068  0.7071068
## 
## $v
##               [,1]       [,2]       [,3]
## [1,] -7.071068e-01 -0.2357023 -0.6666667
## [2,] -7.071068e-01  0.2357023  0.6666667
## [3,]  1.604701e-16 -0.9428090  0.3333333
U = svda$u
S = 0*A
for(k in 1:min(dim(A))) {S[k,k] = svda$d[k]}
V = svda$v
U%*% S %*% t(V)
##      [,1] [,2] [,3]
## [1,]    3    2    2
## [2,]    2    3   -2
zapsmall(V)
##            [,1]       [,2]       [,3]
## [1,] -0.7071068 -0.2357023 -0.6666667
## [2,] -0.7071068  0.2357023  0.6666667
## [3,]  0.0000000 -0.9428090  0.3333333

\(\begin{bmatrix} 2 & 3 & -2 \\ 3 & 2 & 2 \\ \end{bmatrix} = \begin{bmatrix} -\frac{\sqrt2}2 & -\frac{\sqrt2}2 \\ -\frac{\sqrt2}2 & \frac{\sqrt2}2 \\ \end{bmatrix} \begin{bmatrix} 5 & 0 & 0 \\ 0 & 3 & 0 \\ \end{bmatrix} \begin{bmatrix} -\frac{\sqrt2}2 & -\frac{\sqrt2}6 & -\frac23 \\ -\frac{\sqrt2}2 & \frac{\sqrt2}6 & \frac23 \\ 0 & -\frac{2\sqrt2}3 & \frac13 \\ \end{bmatrix}^T\)

4.10

#########
#  4.10 #
#########

U[,1] %*% t(V[,1])
##      [,1] [,2]          [,3]
## [1,]  0.5  0.5 -1.134695e-16
## [2,]  0.5  0.5 -1.134695e-16
zapsmall(5 * U[,1] %*% t(V[,1]))
##      [,1] [,2] [,3]
## [1,]  2.5  2.5    0
## [2,]  2.5  2.5    0

\(A_1 = u_1 v_1^T = 5\ \begin{bmatrix} -\frac{\sqrt2}2 \\ -\frac{\sqrt2}2 \\ \end{bmatrix} \begin{bmatrix} -\frac{\sqrt2}2 \\ -\frac{\sqrt2}2 \\ 0 \\ \end{bmatrix}^T = \begin{bmatrix} \frac52 & \frac52 & 0 \\ \frac52 & \frac52 & 0 \\ \end{bmatrix}= \begin{bmatrix} 2.5 & 2.5 & 0 \\ 2.5 & 2.5 & 0 \\ \end{bmatrix}\)

4.9

#########
#  4.9  #
#########

A = matrix(c(2,-1,2,1), ncol = 2); A
##      [,1] [,2]
## [1,]    2    2
## [2,]   -1    1
svda = svd(A, nu=2, nv = 2); svda
## $d
## [1] 2.828427 1.414214
## 
## $u
##               [,1]         [,2]
## [1,] -1.000000e+00 1.110223e-16
## [2,]  1.110223e-16 1.000000e+00
## 
## $v
##            [,1]       [,2]
## [1,] -0.7071068 -0.7071068
## [2,] -0.7071068  0.7071068
U = svda$u
S = 0*A
for(k in 1:min(dim(A))) {S[k,k] = svda$d[k]}
V = svda$v
U%*% S %*% t(V)
##      [,1] [,2]
## [1,]    2    2
## [2,]   -1    1
zapsmall(U)
##      [,1] [,2]
## [1,]   -1    0
## [2,]    0    1

\(\begin{bmatrix} 2 & 2 \\ -1 & 1 \\ \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} 2 \sqrt2 & 0 \\ 0 & \sqrt2 \\ \end{bmatrix} \begin{bmatrix} -\frac{\sqrt2}2 & -\frac{\sqrt2}2 \\ -\frac{\sqrt2}2 & \frac{\sqrt2}2 \\ \end{bmatrix}^T\)