In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is
[
3
0
0
2
]
{\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}
, while an example of a 3×3 diagonal matrix is
[
6
0
0
0
7
0
0
0
4
]
{\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&7&0\\0&0&4\end{smallmatrix}}\right]}
. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix.
A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.
Homework Statement
Suppose the matrix A , B are diagonalizable and have the same eigenvectors. Show AB=BA
Homework Equations
The Attempt at a Solution
There exists a matrix P s.t. (P^-1)AP=(P^-1)BP
I played around with this, and could not get anywhere..
Perhaps a silly question. I have a vector:
a=[a_1 \ a_2 \ a_3 \ ...\ a_n]^T
that I want to turn into a diagonal matrix. Is there an elegant way to represent this? I thought maybe something like:
a^TI
would do, but it doesn't. I suppose I can use the kronecker delta and subscript...
I have been thinking about this for a week and I simply can't get anywhere.
consider a 4x4 diagonal matrix such that if a11 > a22 > a33 > a44, (0's everywhere else)
then it is similar to another matrix with the same diagonal but 0's in upper triangular part an any numbers in the lower...
diagonal matrix is a matrix with each element is0 excep for elements on the major diagonal.
so, does it mean that
( 2 0 0
0 0 0
0 0 3 )
is not a diagonal matrix??