^{}This may be a silly question, but if I have an 8x3 matrix X, for example, then the rows of this matrix will span R3 (and will be linearly dependent). When we find the solution to:
Xw=t
where t is an 8x1 matrix of t's. Then each row can be represented as...
This is a formula that is seen, for example, in Uniformization:
P_{ij}(t) = P(X(t) = j|X(0) = i)
= \sum^{n=0}_{\infty} P(X(t) = j|X(0) = i, N(t) = n)P(N(t) = n|X(0) = i)
= \sum^{\infty}_{n=0} P^{n}_{ij} \frac{(e^{vt})(vt^n)}{n!}
Where P_{ij}(t) is the transition probability in a...
I'm currently studying experiments where one or more factors are random, i.e. random effects models. In this model a professor explained that the Expected Mean Square calculations for any factor are:
Expected Mean Square (factor) = (lower level error terms) + (term relating to factor)
For...
I have seen in class the following formula used:
P(A | C) = \sum_{B} P(A | B \cap C)*P(B | C)
I don't understand how this formula works? Can anyone help me understand how it can be derived and how I can understand it intuitively? Can a venn diagram be drawn to illustrate this formula?
Homework Statement
This isn't a homework question so I apologize if I'm in the wrong section, but I'm wondering if proofs are 'easy' or 'intuitive' to you. I recently took a linear algebra course in which I was sometimes able to get through the proofs without any trouble but was completely...
Homework Statement
Let A be an m x n matrix with rank(A) = m < n. As far as the eigenvalues of A^{T}A is concerned we can say that...
Homework Equations
The Attempt at a Solution
If eigenvalues exist, then
A^{T}Ax = λx where x ≠ 0.
The only thing I think I can show is that...
I've realized that the matrix given in this problem is an orthogonal matrix, so it rotates the vector x. However, should I be able to see that lamda (and A) perform a rotation if I didn't realize that A was an orthogonal matrix?
Ah, thank you, after fixing the algebra I see the answer is:
λ = cos θ ± isin θ.
In terms of what this means geometrically, it's been a long time since I've done anything with complex numbers, so can you correct the following if it is wrong?
I know that since λ is complex, the...
Homework Statement
Let Q be an orthogonal matrix with an eigenvalue λ_{1} = 1 and let x be an eigenvector belonging to λ_{1}. Show that x is also an eigenvector of Q^{T}.
Homework Equations
Qx = λx where x \neq 0
The Attempt at a Solution
Qx_{1} = x_{1} for some vector x_{1}...
Homework Statement
Show that the matrix
A = [cos θ -sin θ
sin θ cos θ]
will have complex eigenvalues if θ is not a multiple of π. Give a geometric interpretation of this result.
Homework Equations
Ax = λx, so
det(A-λI) = 0
The Attempt at a Solution
In this case...
I have another similar question and I'm hoping I can't get help with both of them. The 2nd question is:
Homework Statement
Determine whether the set of all polynomials in P_{4} (where P_{4} is the set of polynomials with degree less than 4) having at least one real root is a subspace of...
Homework Statement
Determine the null space of the following matrix:
A = [1 1 -1 2
2 2 -3 1
-1 -1 0 -5]
Homework Equations
Ax=0 where x = (x_{1}, x_{2}, x_{3}, x_{4})^{T}
The Attempt at a Solution
If I put the system Ax=0 into augmented form:
1 1 -1 2 | 0
2 2 -3 1 | 0...
Homework Statement
Determine if the set of all singular 2 x 2 matrices are a subspace of R^{2}
Homework Equations
If a, b, c, and d are the entries of a 2 x 2 matrix, then their determinant, ad - bc = 0 if the matrix is singular.
The Attempt at a Solution
I have been doing other...