[Computational Engineering] Lecture 1-1. Matrices

(1) Matrix K

Below are the traits that we can found looking at the given matrix K.

 

• Symmetric: 

1. Easy definition: Original matrix is the same as the matrix which is flipped to the diagonal line of the matrix.
2. Mathematical definition: K = K^T (K^T is a transpose matrix of K)

• Sparse: Mostly Zeros

In the given matrix, we can not say it a sparse matrix; however, considering if we expand the matrix with the same pattern, we can predict the matrix will become sparse.

 

A sparse matrix is important as zeros make calculation very simpler.

 

• Constant diagonal (Toeplitz) = Time invariant = Linear time-invariant

A constant diagonal matrix means that it has all same numbers in a diagonal line. In the case of K, it consists of 2 in the main diagonal line and all in -1 in two diagonal lines which are right next to the main diagonal line.

 

• Invertible issues

Questions.

How do we know the matrix is invertible if the matrix is 100 x 100? (Large enough)

How do we know the whole family of the matrix(Kn) is invertible?

 

Answer.

Row reduce. The computer calculates its invertible matrix by row reducing.

 

In the case of matrix K, by row reducing, we can make a triangular matrix like below.

 

Row reducing matrix K

Note that the determinant of the triangular matrix is the product of pivots. 2⨯3/2⨯4/3⨯5/4. As a result, the determinant of matrix K is 5(≠0) which means K inverse exists.

 

To conclude, we can summarize the existence of an inverse matrix in the upper triangle matrix like below.

 

1. Full set of pivots which is non zero
2. Diagonals with non zero

 

(2) Matrix C

Comparing to matrix K, there are some differences in matrix C.

 

• Symmetric

Just like K, matrix C is also a symmetric matrix. In mathematical definition, C = C^T. (C^T is a transpose matrix of C)

 

• Constant diagonal (Toeplitz)

C is also a Toeplitz matrix with constant diagonal lines.

 

• Invertible issues

However, is C still invertible?

 

We also can try row reduce for matrix C. However, this time we can think differently. Let's imagine we multiply another matrix u with C. If C^-1 exists, the only way to make Cu = 0 is when u =0.

If C^-1 exists, and Cu = 0
C^-1 ⨯ Cu = C^-1 ⨯ 0
I⨯u = 0
u = 0

The problem occurs here. In the case of C, there is another matrix u that can make C to 0. Which means that c^-1 can't exist.

u(≠0) that makes Cu = 0

 

• Periodic

This kind of inverse matrix issue occurred as the matrix is was a periodic matrix. The component of the matrix rolls over to another component.

 

(3) Physical expressions of Matrix K and C

It is interesting that we can express both matrices in a physical 3d model. For matrix K, the model can be defined as four components connected one by one with the boundary condition of the fixed end. The whole system will be fixed and can't move. (u=0) In other words, displacement is zero.

 

However, for matrix C, as it was a periodic(circular, cyclic) matrix, we can imagine it as constructing a ring-like shape., resulting in the whole system can turn around, with equal displacement. (u≠0) In other words, the slope is zero.

 

Matrix K = Both fixed end model

Matrix C = Circular model

(4) Modifications of matrix K

Matrix had a fixed end for both ends. However, we also can think about two different variations of these end types. The first one will be a free-fixed type, and the other one will be a free-free type.

 

1. Free - fixed type

Matrix T = Free - fixed model

 

• Symmetric: T is not a symmetric matrix anymore.
• Constant diagonal (Toeplitz): T is not a constant diagonal matrix anymore. (1,1) is different from the rest of the diagonal components.
• Invertible issues: T is invertible. The whole system won't move.

 

2. Free - free type

Matrix B = Free - free model

 

• Symmetric: B is not a symmetric matrix anymore.
• Constant diagonal (Toeplitz): T is not a constant diagonal matrix anymore. (1,1) and (4,4) is different from the rest of the diagonal components.
• Invertible issues: B is not invertible. The whole system can move freely. Also has u≠0 which makes Bu = 0.

(5) Summary

There are four different types of matrices we covered. To summarize and give names to each matrix,

 

Matrix K and T = Positive definite matrix

Matrix C and B = Positive semi-definite matrix

---

* This post is a personal study archive of Professor Gilbert Strang's 18.085 Computational Science and Engineering course at MIT Open Course Ware.
* Please refer to the link below for lecture information.

 

https://youtu.be/C6 pEBqqYnWIocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/