## Why do the eigenvalues of a triangular matrix lie on its diagonal?

Suppose is a triangular matrix, so is invertible. Now with eigenvalues finding process, we can only shift each ‘s diagonal entry for a to make non-invertible. And the only possible way is to shift each ‘s diagonal entry for a that is also an ‘s diagonal entry, so after that one diagonal entry of will equal […]

## Pictorial proof that Reflection(v) = 2*Projection(v) – Identity(v)

Hyperplane here is a 1-D line, I don’t know if this visual proof holds for n-dimensional space haha…

## Proof that unless projection matrix P = I, P is singular

We have: , first we need to prove that : Set , hence each column vector of is a linear combination of column vectors of , so . Here is an by matrix while is an by matrix and column vectors of must be linearly indepedent (so ), hence . If then it’s trivial to show that is […]

## Learning From Data – A Short Course: Problem 7.1

Page 43 Implement the decision function below using a 3-layer perceptron. First I’ll construct a rectangle like this: It’s easy to see how: Consider the four lines , , , and what we want is the hypothesis . The corresponding MLP: Next I’ll try to construct a cooler shape: Now consider the three lines , and […]

## Learning From Data – A Short Course: Exercise 8.17

Page 45 Show that is an upper bound on the , where  is the classification error. We consider the error and on data point . (correct classification):     (wrong classification): We have:     Hence:     So the statement follows.

## Learning From Data – A Short Course: Exercise 8.12

Page 29 If all the data is from one class, then for . (a) What is ? (b) What is ? From (8.23) we have . As all the data is from one class, we also have: . Hence: