Intro
Anyone who has ever tried adding, subtracting, or multiplying by zero knows that it is a very convenient number to do arithmetic with.
These days, when most calculations are made by computers, this simplicity has carried over to the digital realm. This should be no surprise since computers are built by humans after all.
Computers often performs calculations on matrices, and it turns out that special types of matrices with many of their elements being zero make calculations both faster and more accurate.
Larry Page and Sergey Brin, the founders of Google, knew everything about the workings of computer arithmetic and how to optimize it.
This allowed for the revolution of the search engine market that Google brought about, increasing both the rate of relevant hits on the web per unit time manifold as compared to their competitors.
Concept
There are a couple special types of matrices which are particularly interesting to us. They all tend to be square shaped and usually contain a lot of zeros.
Diagonal matrices have all elements zero except for the ones along the main diagonal. A special case of a diagonal matrix is the identity matrix .
Example:
In addition, triangular matrices can also have non-zero elements above the main diagonal, making it upper triangular, or below it to take the shape of a lower triangular matrix:
Another special type of matrices, symmetric ones, has its elements above the main diagonal as the mirror image of the ones below it.
Example:
Math
The reason these particular forms are of interest to us is that the math tend to simplify significantly.
Adding, subtracting, or multiplying two matrices that are both either diagonal or triangular always produce a matrix of that same form.
Hence, we can right away insert a bunch of zeros in the resulting matrix without wasting time on going through unnecessary calculations.
Example:
Symmetric matrices have beneficial characteristics as well, one of them being that taking the transpose gives you back the same matrix.
Diagonal matrices
A square -matrix whose elements are all 0 except for the diagonal elements is called a diagonal matrix. The general form therefore follows:
where are real numbers. The diagonal matrix is only invertible if all diagonal elements are non-zero, otherwise the columns will be linearly dependent. If this is the case,
Feel free to check this by confirming that:
After that confirmation, it becomes relatively easy to multiply by itself times resulting in:
which applies to all integers, positive and negative, .
Triangular matrices
A triangular matrix is a square -matrix that has all the elements above or below the diagonal being 0, whereupon it is then called an upper traingular or lower triangular matrix. They follow the forms:
In addition, should all diagonal elements consist of zeros, the matrices are called strictly upper triangular and strictly lower triangular matricies, respectively. They follow the formats:
The following theorem is useful to know about triangular matrices:
Properties of triangular matrices.
The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
The product of two upper triangular matrices is upper triangular, while the product of two lower triangular matrices is lower triangular.
A triangular matrix is invertible if, and only if, all diagonal elements are non-zero.
The inverse of an invertible upper triangular matrix is upper triangular, while the inverse of an invertible lower triangular matrix is lower triangular.
Symmetric and skew-symmetric matrices
A square matrix is symmetric if and skew-symmetric if . Remember that the transpose can be seen as a reflection of the elements around the diagonal. Examples of symmetric matrices are:
and examples of skewed-symmetric matrices would be:
Note that the diagonal of a skewed-symmetric matrix must be zero for the requirement to hold, as the diagonal is fixed and is not reflected in the transposition.
If and are symmetric matrices with the same dimensions, and if is a scalar, then:
and are symmetric
and are symmetric
is symmetric
The product of and is not symmetric, because:
For to be equivalent to , it must also be equivalent to according to the above statements, i.e. we set the requirement that:
Since we know that matrix multiplication does not commutate, it therefore holds that the product between and is symmetric about, and only about, .
An additional useful theorem is:
If is an invertible, symmetric, matrix, then is also symmetric.
Assume that is invertible and symmetric. Then we have:
