**3**

The algebra of square matrices

**3.1 The summation convention**

In our first chapter we showed how a system of *n* linear equations in *n* unknowns could be written compactly as

In 1916, Einstein wrote to a friend (see, for example, [26])

I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice . . .

Although Einstein wrote with his tongue in his cheek, the *Einstein summation convention* has proved very useful. When we use the summation convention with *i*, *j*, . . . running from 1 to *n*, then we must observe the following rules.

(1) Whenever the ...

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