Basis for the set of diagonal matrices
Let A be the set of all diagonal matrices. How can we calculate it's basis and dimension? I am not really good at it. Can you help me with this, so thus I will have some picture.
$\endgroup$ 23 Answers
$\begingroup$Hint: Do you know a basis for the vector space of all matrices of a fixed size? The basis for $n \times n$ matrices is the set of matrices having exactly one entry equal to $1$ and all the rest equal to zero. There are $n^2$ such matrices. Do you see why? Once you understand this, consider restricting your attention to the diagonal matrices in this basis.
$\endgroup$ $\begingroup$Here's a good hint for these problems: intuitively, the dimension of a vector space is the number of "coordinates" (i.e. scalars) required to describe each point uniquely.
Surely, then, it makes sense to describe the $n \times n$ diagonal matrices using $n$ coordinates. Indeed, the "common-sense" approach corresponds to using the basis $$ \pmatrix{1\\&0\\&&0\\ && & \ddots \\ && & & 0}, \quad \pmatrix{0\\&1\\&&0\\&&&\ddots \\&&&&0}, \dots $$
$\endgroup$ $\begingroup$Hint:
$$ \begin{bmatrix} a_1&0&0&\cdots &0\\ 0&a_2&0&\cdots&0\\ 0&0&a_3 &\cdots &0\\ \cdots\\ 0&0&0&\cdots &a_n \end{bmatrix}= a_1 \begin{bmatrix} 1&0&0&\cdots &0\\ 0&0&0&\cdots&0\\ 0&0&0 &\cdots &0\\ \cdots\\ 0&0&0&\cdots &0 \end{bmatrix}+ a_2 \begin{bmatrix} 0&0&0&\cdots &0\\ 0&1&0&\cdots&0\\ 0&0&0 &\cdots &0\\ \cdots\\ 0&0&0&\cdots &0 \end{bmatrix}+ a_3 \begin{bmatrix} 0&0&0&\cdots &0\\ 0&0&0&\cdots&0\\ 0&0&1 &\cdots &0\\ \cdots\\ 0&0&0&\cdots &0 \end{bmatrix}+ \cdots+ a_n \begin{bmatrix} 0&0&0&\cdots &0\\ 0&0&0&\cdots&0\\ 0&0&0 &\cdots &0\\ \cdots\\ 0&0&0&\cdots &1 \end{bmatrix} $$
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