Question

symmetric second-order tensor. Such axes are called the principal axes of the tensor. For this case, the basis vectors are actually the unit principal directions l"i". nan, nay. and it can be shown that with respect to principal axes the tensor reduces to the diagonal form Note that the fundamental invariants defined by relations (1.6.3) can be expressed in terms of the principal values as The eigenvalues have important extremal properties. If we arbitrarily rank the priacipal values such that λǐ > a2 > as, then λ 1 will be the largest of all possible diagonal elemen(s. smallest diagonal element possible. This theory is applied in elasticity as yeyock the largest stress o strain components in an elastic solid. λ3 will be th EXAMPLE 1.3: PRINCIPAL VALUE PROBLEM Determine the invariants and principal values and directions of the following symmetric second- order tensor 2 0 0 04-3 The invariants follow from relations (1.6.3) 6-25-625 03 14-310 -3 lll-lo 3-41-2(-9-16)--50 4-3 0 The characteristic equation then becomes (2-2) (F-25) 0 Thus, for this case all principal values are distinct.

how can we solve this type problems we dont want specific solution. we want to use as A input everytime different matrix. three axis have each rotation degree

Answer 1

I am a data communications technician who programs in his spare time.
My question is this.  What is the formula to solve for x,y,z - given
a current x,y,z coordinate (i.e. -60,-60,-60) and an angle
(i.e. 90 degrees) and a plane to rotate about (i.e. z)?

I am currently taking x,y and solving for its angle (with ATN) and radius, 
adding to the angle (i.e. 90 degrees), then using sine and cosine to 
re-get an x,y coordinate. (Does this make sense?)

In my example I would get (60,-60,-60 x,y,z) using my method.
But it's a tedious method.

Why am I doing this?  I am rotating objects in 3 dimensions.

It's been years since I've taken math classes.  I've been trying to
find some good resources on the internet and referred to some old
books I have

Matrices are a kind of mathematical object.  The theory of matrices is
complicated (being essentially the subject of a semester-long college math
class called "linear algebra"), but the practical deal, as far as rotation
is concerned, is pretty simple.

A matrix is usually represented as a rectangular array of values (usually 
numbers); you usually work with matrices of a particular size, such as 
3x3 (first being rows, second being columns).

Matrices can be multiplied; any mXn matrix can be multiplied by an nXp
matrix to yield an mXp matrix.  Note that this multiplication is not
commutative; if A and B are matrices, AB is not necessarily the same as BA.
One may even exist without the other existing.

Matrices are multiplied like this:

If A is a mXn matrix like
   a[1,1]   a[1,2] ... a[1,n]
   a[2,1]   a[2,2] ... a[2,n]
   .
   .
   a[m,1]   a[m,2] ... a[m,n]

(note the order: row, then column)

and B is a nXp matrix, equivalently represented (with b's)
Let C=AB; C is a mXp matrix, likewise with c's.
So for any i,j,
c[i,j] = a[i,1]*b[1,j] + a[i,2]*b[2,j] + ... + a[i,n]*b[n,j]
You dig?  If you have questions, write back with them.

A vector (like the coordinates (-60,-60,-60) you gave) can be represented
by a matrix; a 3-vector can either be a 3x1 matrix or a 1x3 one.

When you multiply a matrix and a vector (of compatible sizes, on the
appropriate side) you get another vector, which is the first vector after
being  transformed in various ways.