Diagonalising Matrices

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A diagonal matrix is a square matrix where every element except the leading diagonal is zero.

Since most of the elements are zero, certain calculations on the matrix are a lot easier than a general matrix. For example, the determinant of a large, general matrix involves a large number of multiplications, but the determinant of a diagonal matrix is simply the product of the diagonal elements.

Of course, most matrices are not diagonal. However the process of diagonalisation often allows us to find a diagonal matrix that shares some of the properties of the original matrix. We can then perform calculations a lot more easily on the diagonal matrix. The results can then be applied back to the original matrix.

We say that the diagonalised matrix is similar to the original matrix (a term that we will define below), which allows certain calculations to be shared by both matrices.

In this article, we will look at the properties of diagonal matrices, and define what we mean by similar and diagonalised matrices. We will then look at several practical methods of diagonalising a matrix.

Diagonal matrices

Here is a general matrix. We will use a three-by-three matrix as an example:

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