start by finding the eigenvalues. That is, find the values L for which the matrix
(A - I*L) =
[i-L...0......0]
[0....-L......i]
[0.....i......-L]
has a determinant of zero ("I" above refers to the identity matrix).
Going through the math, you should find the eigenvalues of this matrix to i, i, and -i (we count i twice because it is a root of multiplicity 2). Now to find the eigenvectors corresponding to an eigenvector L, we need to find the non-zero vectors v so that
A*v = L*v
which is the same as saying
(A - L*I)*v = 0
In the case of L = i, we need to find the non-zero vectors v for which
A - i*I =
[0...0....0]
[0...-i.....i] * V = 0
[0....i....-i]
Or, phrased differently, we need to find "a basis for the kernel" of (A - i*I).
As you may check, the following two are solutions to the above
v_1 =
[1]
[0]
[0]
v_2 =
[0]
[1]
[1]
Doing the same for L = -i, we get the non-zero vector
v_3 =
[0]
[1]
[-1]
Thus, v_1, v_2, and v_3 above are a basis of eigenvectors.
Hope that helps, let me know if anything is unclear.
Also, a good way to find the answers to these things quickly (if you need to check):
Been a protracted time on condition that i've got performed this, in all possibility incorrect. yet ill attempt. a million.) If the boat is travelling at a cost of 4mph, however the river is offering resistance equivalent to 3mph, than the consequent speed often is the forward cost (the intial velocity) takeaway the resistance. So subsequently: 4 - 3 = a million. 2.) the magnitude will the full stress being utilized to the boat = seven hundred kilos ( i think of - may be incorrect). The process the consequent stress would be in the destructive XY airplane, with the arror pointing in the destructive x course, yet 3 quarters down the destructive y airplane. e.g. in case you probably did a standard -4x-4 xy plot, placed the arrow at (-4, -3) with the arrow head pointing in direction of what could be -5x (if it is clever). unsure if it is stable, little doubt somebody will tell me it relatively is incorrect, yet magnitude is many times entire stress being utilized to an merchandise, the ensuing course often is the consequences of the two the forces, draw it out like above and you will even see whats occurring,. The x cost is extra destructive than the y simply by fact the stress utilized is enhanced in that axis - to a ratio 4:3, for this reason x y plot of (-4, -3)
Comments
start by finding the eigenvalues. That is, find the values L for which the matrix
(A - I*L) =
[i-L...0......0]
[0....-L......i]
[0.....i......-L]
has a determinant of zero ("I" above refers to the identity matrix).
Going through the math, you should find the eigenvalues of this matrix to i, i, and -i (we count i twice because it is a root of multiplicity 2). Now to find the eigenvectors corresponding to an eigenvector L, we need to find the non-zero vectors v so that
A*v = L*v
which is the same as saying
(A - L*I)*v = 0
In the case of L = i, we need to find the non-zero vectors v for which
A - i*I =
[0...0....0]
[0...-i.....i] * V = 0
[0....i....-i]
Or, phrased differently, we need to find "a basis for the kernel" of (A - i*I).
As you may check, the following two are solutions to the above
v_1 =
[1]
[0]
[0]
v_2 =
[0]
[1]
[1]
Doing the same for L = -i, we get the non-zero vector
v_3 =
[0]
[1]
[-1]
Thus, v_1, v_2, and v_3 above are a basis of eigenvectors.
Hope that helps, let me know if anything is unclear.
Also, a good way to find the answers to these things quickly (if you need to check):
http://www.wolframalpha.com/input/?i=eigenvalues+o...
Been a protracted time on condition that i've got performed this, in all possibility incorrect. yet ill attempt. a million.) If the boat is travelling at a cost of 4mph, however the river is offering resistance equivalent to 3mph, than the consequent speed often is the forward cost (the intial velocity) takeaway the resistance. So subsequently: 4 - 3 = a million. 2.) the magnitude will the full stress being utilized to the boat = seven hundred kilos ( i think of - may be incorrect). The process the consequent stress would be in the destructive XY airplane, with the arror pointing in the destructive x course, yet 3 quarters down the destructive y airplane. e.g. in case you probably did a standard -4x-4 xy plot, placed the arrow at (-4, -3) with the arrow head pointing in direction of what could be -5x (if it is clever). unsure if it is stable, little doubt somebody will tell me it relatively is incorrect, yet magnitude is many times entire stress being utilized to an merchandise, the ensuing course often is the consequences of the two the forces, draw it out like above and you will even see whats occurring,. The x cost is extra destructive than the y simply by fact the stress utilized is enhanced in that axis - to a ratio 4:3, for this reason x y plot of (-4, -3)