Matrix:
[1 2; 3 4] ans = 1 2 3 4
Row vector addition:
[1 2] + [2 3] ans = 3 5
Column vector multiplication and addition:
v = [2 3 4]' ; w = [1 1 1]' ; u = 2 * v + 3 * w u = 7 9 11
>> eye(2) ans = 1 0 0 1
>> ones(2,2) ans = 1 1 1 1
>> 6*eye(5)-ones(5,5) ans = 5 -1 -1 -1 -1 -1 5 -1 -1 -1 -1 -1 5 -1 -1 -1 -1 -1 5 -1 -1 -1 -1 -1 5
>> b = rand(4,1) b = 0.8147 0.9058 0.1270 0.9134
The dot product is a row vector times a column vector.
Dot product:
[1 2] * [3 4]' ans = 11
The length of vector:
v = [1 2 2] ; norm(v) ans = 3
The vector length is the same as sqrt(v*v):
v = [1 2 2] ; sqrt(v * v') ans = 3
Cosine between two vectors:
v = [1 2] ; w = [3 4]; cosine = v * w' / (norm(v) * norm(w)) cosine = 0.9839
Angle (in radians) between two unit-length vectors:
i = [1 0] ; j = [0 1] ; cosine = i * j' ; angle = acos(cosine) angle = 1.5708
Special matrices:
>> pascal(4) // 4 x 4 symmetric Pascal matrix ans = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 >> inv(pascal(4)) ans = 4.0000 -6.0000 4.0000 -1.0000 -6.0000 14.0000 -11.0000 3.0000 4.0000 -11.0000 10.0000 -3.0000 -1.0000 3.0000 -3.0000 1.0000 >> L = abs(pascal(4,1)) // Pascal's lower triangular ans = 1 0 0 0 1 -1 0 0 1 -2 1 0 1 -3 3 -1 >> L * L' // == pascal(4) ans = 1 1 1 1 1 2 3 4 1 3 6 10 1 4 10 20 >> inv(L') * inv(L) // == inv(pascal(4)) ans = 4 -6 4 -1 -6 14 -11 3 4 -11 10 -3 -1 3 -3 1 >> hilb(6) // the approximation of a Hilbert matrix (fractions are rounded off) ans = 1.0000 0.5000 0.3333 0.2500 0.2000 0.1667 0.5000 0.3333 0.2500 0.2000 0.1667 0.1429 0.3333 0.2500 0.2000 0.1667 0.1429 0.1250 0.2500 0.2000 0.1667 0.1429 0.1250 0.1111 0.2000 0.1667 0.1429 0.1250 0.1111 0.1000 0.1667 0.1429 0.1250 0.1111 0.1000 0.0909 >> inv(hilb(6)) // the approximated inverse of the Hilbert matrix ans = 1.0e+06 * 0.0000 -0.0006 0.0034 -0.0076 0.0076 -0.0028 -0.0006 0.0147 -0.0882 0.2117 -0.2205 0.0832 0.0034 -0.0882 0.5645 -1.4112 1.5120 -0.5821 -0.0076 0.2117 -1.4112 3.6288 -3.9690 1.5523 0.0076 -0.2205 1.5120 -3.9690 4.4100 -1.7464 -0.0028 0.0832 -0.5821 1.5523 -1.7464 0.6985 >> invhilb(6) // the exact inverse of the Hilbert matrix ans = 36 -630 3360 -7560 7560 -2772 -630 14700 -88200 211680 -220500 83160 3360 -88200 564480 -1411200 1512000 -582120 -7560 211680 -1411200 3628800 -3969000 1552320 7560 -220500 1512000 -3969000 4410000 -1746360 -2772 83160 -582120 1552320 -1746360 698544