by Matthew Leitch, 25 October 2001

Some things are really tough to understand at all. Normally this is partly the subject itself and mostly the way it is explained. Often it is hard because the source assumes more knowledge than you actually have. What makes something hard is that ideas you need to learn are explained using terms/ideas you don't already know. You have to be aware of this and backtrack where possible. You also have to learn what you can, notice the gaps, and keep searching until you fill them in.

Using your memory effectively is the key to reaching understanding. You must learn things as you go through or you will never understand.

**Example: A mathematical definition.** For example, a dictionary of mathematics says that a **diagonally dominant matrix** is "a SYMMETRIC MATRIX in which every element in the MAIN DIAGONAL is larger than the sum of the absolute values of the remaining elements of its row." I already know what a matrix is, but observe that:

- [noticing the label] the new term is "diagonally dominant matrix";
- [noticing the concept] so it's a type of matrix;
- [noticing the label] "dominant" not leading, or some other word of similar meaning;
- [noticing the label] diagonally - not some other direction;
- I do not know what a symmetric matrix is;
- [noticing the label] however there is a term "symmetric matrix";
- [noticing the label] the name is "symmetric" matrix, not "symmetrical";
- I do not know what a main diagonal is; and
- [noticing the label] however there is a term "main diagonal";
- [noticing the label] the name is "main", not "leading" or some other word with similar meaning;
- the rule is rather complex and I will have to approach it in stages, building up chunks to reach the full concept.

The dictionary says that a symmetric matrix is "a square matrix whose entries are symmetric around its main diagonal" and it gives an example. I already know about square matrices and about diagonals. From the example I can notice that:

- [link label to concept] the "main diagonal" must the one from top left to bottom right;
- [reason] that seems right as we normally read from top left to bottom right;
- [reason] and hence the "diagonal" in "diagonally dominant";
- symmetric means the values are equal to their reflection on the other side of the diagonal;
- a symmetric matrix has to be square;
- [reason] which seems right since otherwise some numbers would have no reflection to match;
- [deduction] you can flip a symmetric matrix about its main diagonal and it is still the same;
- [deduction] each row has an equal reflected column and they intersect on the main diagonal.

Now back to that original definition.

- the rule is that something must be true of every element of the main diagonal;
- that something concerns the value of the element on the main diagonal for each row;
- it must be larger than some other number;
- notice it is larger than i.e. greater than, not greater than or equal;
- [reason] hence the "dominant" in "diagonally dominant matrix";
- the thing it must be larger than is a sum;
- the sum of the other elements on the row;
- notice it doesn't include the element itself;
- [reason] that figures because if it included itself it would be an odd sense of "dominant";
- and it uses the absolute value of each element;
- though not the absolute value of the element on the main diagonal;
- so the number for the comparison is the sum of the absolute values of the other elements of the row [a key chunk to form];
- [deduction] that number will always be non-negative;
- [reason] because it is the sum of absolute numbers:
- non-negative not positive - it could be zero;
- [deduction] so the elements on the main diagonal must all be non-negative;
- [deduction] the definition would be the same if it referred to columns instead of rows.

The deductions are important as they use and so reinforce my new memories, create useful redundancy, and give me a headstart in reasoning about diagnonally dominant matrices. Now if I have to learn something more about them I have more chunks and facts to build with and my inferences will be quicker and more numerous.

This example had a happy ending as I was able to go back and find all the explanations I needed. Often this is not possible and learning is, unavoidably, damaged. Well written explanations give you the elements you need in the right order so there is no need to go back. School text books often achieve this. I hope this web page also does.

© 2001 Matthew Leitch