Matrix review and svd preview
2006.06.27 mjf
Contents
Setup A and B
syms a11 a12 a13 a14 real; syms a21 a22 a23 a24 real; syms a31 a32 a33 a34 real; syms a41 a42 a43 a44 real; syms b11 b12 b13 b14 real; syms b21 b22 b23 b24 real; syms b31 b32 b33 b34 real; syms b41 b42 b43 b44 real; A = [a11 a12 a13 a14; a21 a22 a23 a24; a31 a32 a33 a34; a41 a42 a43 a44] B = [b11 b12 b13 b14; b21 b22 b23 b24; b31 b32 b33 b34; b41 b42 b43 b44]
A = [ a11, a12, a13, a14] [ a21, a22, a23, a24] [ a31, a32, a33, a34] [ a41, a42, a43, a44] B = [ b11, b12, b13, b14] [ b21, b22, b23, b24] [ b31, b32, b33, b34] [ b41, b42, b43, b44]
A*B
A*B
ans = [ a11*b11+a12*b21+a13*b31+a14*b41, a11*b12+a12*b22+a13*b32+a14*b42, a11*b13+a12*b23+a13*b33+a14*b43, a11*b14+a12*b24+a13*b34+a14*b44] [ a21*b11+a22*b21+a23*b31+a24*b41, a21*b12+a22*b22+a23*b32+a24*b42, a21*b13+a22*b23+a23*b33+a24*b43, a21*b14+a22*b24+a23*b34+a24*b44] [ a31*b11+a32*b21+a33*b31+a34*b41, a31*b12+a32*b22+a33*b32+a34*b42, a31*b13+a32*b23+a33*b33+a34*b43, a31*b14+a32*b24+a33*b34+a34*b44] [ a41*b11+a42*b21+a43*b31+a44*b41, a41*b12+a42*b22+a43*b32+a44*b42, a41*b13+a42*b23+a43*b33+a44*b43, a41*b14+a42*b24+a43*b34+a44*b44]
A*B'
A*B'
ans = [ a11*b11+a12*b12+a13*b13+a14*b14, a11*b21+a12*b22+a13*b23+a14*b24, a11*b31+a12*b32+a13*b33+a14*b34, a11*b41+a12*b42+a13*b43+a14*b44] [ a21*b11+a22*b12+a23*b13+a24*b14, a21*b21+a22*b22+a23*b23+a24*b24, a21*b31+a22*b32+a23*b33+a24*b34, a21*b41+a22*b42+a23*b43+a24*b44] [ a31*b11+a32*b12+a33*b13+a34*b14, a31*b21+a32*b22+a33*b23+a34*b24, a31*b31+a32*b32+a33*b33+a34*b34, a31*b41+a32*b42+a33*b43+a34*b44] [ a41*b11+a42*b12+a43*b13+a44*b14, a41*b21+a42*b22+a43*b23+a44*b24, a41*b31+a42*b32+a43*b33+a44*b34, a41*b41+a42*b42+a43*b43+a44*b44]
Set up L similar to singular values
L = diag([5 4 3 2])
L =
5 0 0 0
0 4 0 0
0 0 3 0
0 0 0 2
A*L*B'
Notice the scaling. First singular value scales A11:a41 the first column of A, and b11:b41 the first column of B (first row of B').
M=A*L*B'
M = [ 5*a11*b11+4*a12*b12+3*a13*b13+2*a14*b14, 5*a11*b21+4*a12*b22+3*a13*b23+2*a14*b24, 5*a11*b31+4*a12*b32+3*a13*b33+2*a14*b34, 5*a11*b41+4*a12*b42+3*a13*b43+2*a14*b44] [ 5*a21*b11+4*a22*b12+3*a23*b13+2*a24*b14, 5*a21*b21+4*a22*b22+3*a23*b23+2*a24*b24, 5*a21*b31+4*a22*b32+3*a23*b33+2*a24*b34, 5*a21*b41+4*a22*b42+3*a23*b43+2*a24*b44] [ 5*a31*b11+4*a32*b12+3*a33*b13+2*a34*b14, 5*a31*b21+4*a32*b22+3*a33*b23+2*a34*b24, 5*a31*b31+4*a32*b32+3*a33*b33+2*a34*b34, 5*a31*b41+4*a32*b42+3*a33*b43+2*a34*b44] [ 5*a41*b11+4*a42*b12+3*a43*b13+2*a44*b14, 5*a41*b21+4*a42*b22+3*a43*b23+2*a44*b24, 5*a41*b31+4*a42*b32+3*a43*b33+2*a44*b34, 5*a41*b41+4*a42*b42+3*a43*b43+2*a44*b44]
Now the effect on a vector x (Mx = y)
syms x1 x2 x3 x4 real; x = [x1 x2 x3 x4]' y = M*x
x = x1 x2 x3 x4 y = (5*a11*b11+4*a12*b12+3*a13*b13+2*a14*b14)*x1+(5*a11*b21+4*a12*b22+3*a13*b23+2*a14*b24)*x2+(5*a11*b31+4*a12*b32+3*a13*b33+2*a14*b34)*x3+(5*a11*b41+4*a12*b42+3*a13*b43+2*a14*b44)*x4 (5*a21*b11+4*a22*b12+3*a23*b13+2*a24*b14)*x1+(5*a21*b21+4*a22*b22+3*a23*b23+2*a24*b24)*x2+(5*a21*b31+4*a22*b32+3*a23*b33+2*a24*b34)*x3+(5*a21*b41+4*a22*b42+3*a23*b43+2*a24*b44)*x4 (5*a31*b11+4*a32*b12+3*a33*b13+2*a34*b14)*x1+(5*a31*b21+4*a32*b22+3*a33*b23+2*a34*b24)*x2+(5*a31*b31+4*a32*b32+3*a33*b33+2*a34*b34)*x3+(5*a31*b41+4*a32*b42+3*a33*b43+2*a34*b44)*x4 (5*a41*b11+4*a42*b12+3*a43*b13+2*a44*b14)*x1+(5*a41*b21+4*a42*b22+3*a43*b23+2*a44*b24)*x2+(5*a41*b31+4*a42*b32+3*a43*b33+2*a44*b34)*x3+(5*a41*b41+4*a42*b42+3*a43*b43+2*a44*b44)*x4