FANDOM


In linear algebra, a diagonal matrix is a particular type of square matrix. The main-diagonal may contain any real values, but the elements that do not coincide with the main-diagonal must all be zeros.

In particular, given any ring R with additive identity given by 0, a matrix A \in R^{n \times n} (an n-by-n matrix whose entries are elements of R) is said to be a diagonal matrix if for any j and k such that j \ne k, A_{j,k}=0, where A_{j,k} denotes the j,k-th entry.

An identity matrix is a special case of the diagonal matrix.

A matrix is considered diagonalizable if it is similar to a diagonal matrix. The process of finding the similar diagonal matrix is called diagonalization.

Diagonal matrices are very useful computationally, since

\begin{bmatrix}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c\end{bmatrix}^n
=
\begin{bmatrix}
a^n & 0 & 0\\
0 & b^n & 0\\
0 & 0 & c^n \end{bmatrix}
\begin{bmatrix}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c\end{bmatrix}
\begin{bmatrix}
x & 0 & 0\\
0 & y & 0\\
0 & 0 & z \end{bmatrix}
=
\begin{bmatrix}
ax & 0 & 0\\
0 & by & 0\\
0 & 0 & cz \end{bmatrix}
\begin{bmatrix}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c\end{bmatrix} +
\begin{bmatrix}
x & 0 & 0\\
0 & y & 0\\
0 & 0 & z \end{bmatrix}
=
\begin{bmatrix}
a+x & 0 & 0\\
0 & b+y & 0\\
0 & 0 & c+z \end{bmatrix}
\begin{bmatrix}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c\end{bmatrix}^{-1}
=
\begin{bmatrix}
\tfrac{1}{a} & 0 & 0\\
0 & \tfrac{1}{b} & 0\\
0 & 0 & \tfrac{1}{c} \end{bmatrix}
\begin{vmatrix}
a & 0 & 0\\
0 & b & 0\\
0 & 0 & c\end{vmatrix}
= abc

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