Diagonal matrix

ImprimirCitar

In linear algebra, one Diagonal matrix is a matrix whose elements outside the main diagonal are all zero; the term usually refers to square matrices. An example of a diagonal matrix of size ${displaystyle 2times 2}$ That's it.

{displaystyle {begin{bmatrix}3&0\0&2end{bmatrix}}}

while an example of a size matrix ${displaystyle 3times 3}$ That's it.

{displaystyle {begin{bmatrix}6&0&0\0&7&0\0&0&4end{bmatrix}}}

The identity matrix of any size or any multiple of it (a scalar matrix) is a diagonal matrix.

Definition

The matrix ${displaystyle D=(d_{i,j})}$ with $n$ columns and $n$ it is diagonal if

{displaystyle d_{i,j}=0;{mbox{si}};ineq jquad forall ;i,jin {1,2,dotsn}}

The elements of the main diagonal of the matrix $D$ can take any value.

Every diagonal matrix is also a symmetric, triangular (upper and lower) and (if the entries come from the R or C) body matrix normal.

Vector operations

Multiplying a vector by a diagonal matrix involves multiplying each element of the vector by the corresponding element of the diagonal. Given a diagonal matrix ${displaystyle D=operatorname {diag} (a_{1},dotsa_{n})}$ and a vector ${displaystyle mathbf {v} ={begin{bmatrix}x_{1}&cdots &x_{n}end{bmatrix}}^{T}}$ the product is:

{displaystyle Dmathbf {v} =operatorname {diag} (a_{1},dotsa_{n}){begin{bmatrix}x_{1}\vdots \x_{n}end{bmatrix}}={begin{bmatrix}a_{1}\&ddots \&&a_{n}end{bmatrix}}{begin{bmatrix}x_{1}\vdots \x_{n}end{bmatrix}}={begin{bmatrix}a_{1}x_{1}\vdots \a_{n}x_{n}end{bmatrix}}}

Matrix operations

The sum and multiplication operations between diagonal matrices are very simple. Consider two diagonal matrices of the same size ${displaystyle D=operatorname {diag} (a_{1},dotsa_{n})}$ and ${displaystyle B=operatorname {diag} (b_{1},dotsb_{n})}$ .

For the sum of diagonal matrices we have

{displaystyle {begin{aligned}D+B&=operatorname {diag} (a_{1},dotsa_{n})+operatorname {diag} (b_{1},dotsb_{n})\&={begin{bmatrix}a_{1}\&ddots \&&a_{n}end{bmatrix}}+{begin{bmatrix}b_{1}\&ddots \&&b_{n}end{bmatrix}}\&={begin{bmatrix}a_{1}+b_{1}\&ddots \&&b_{n}+b_{n}end{bmatrix}}\&=operatorname {diag} (a_{1}+b_{1},dotsa_{n}+b_{n})end{aligned}}}

y for the product of matrices,

{displaystyle {begin{aligned}DB&=operatorname {diag} (a_{1},dotsa_{n})cdot operatorname {diag} (b_{1},dotsb_{n})\&={begin{bmatrix}a_{1}\&ddots \&&a_{n}end{bmatrix}}{begin{bmatrix}b_{1}\&ddots \&&b_{n}end{bmatrix}}\&={begin{bmatrix}a_{1}b_{1}\&ddots \&&a_{n}b_{n}end{bmatrix}}\&=operatorname {diag} (a_{1}b_{1},dotsa_{n}b_{n})end{aligned}}}

The diagonal matrix ${displaystyle D=operatorname {diag} (a_{1},dotsa_{n})}$ is invertible if and only if the tickets ${displaystyle a_{1},dotsa_{n}}$ They're all different than 0. In this case, you have

{displaystyle D^{-1}=operatorname {diag} (a_{1},dotsa_{n})^{-1}=operatorname {diag} (a_{1}^{-1},dotsa_{n}^{-1})}

In particular, diagonal matrices form a ring of the matrices $ntimes n$ .

Multiply the matrix $A$ for Left with ${displaystyle operatorname {diag} (a_{1},dotsa_{n})}$ equivalent to multiplying $i$ -thousand row $A$ for ${displaystyle a_{i}}$ for everything $i$ . Multiply the matrix $A$ for Right. with ${displaystyle operatorname {diag} (a_{1},dotsa_{n})}$ equivalent to multiplying $i$ -thousand column $A$ for ${displaystyle a_{i}}$ for everything $i$ .

Properties

The determining factor ${displaystyle operatorname {diag} (a_{1},dotsa_{n})}$ equals the product ${displaystyle a_{1}cdots a_{n}}$ .
The attachment of a diagonal matrix is also a diagonal matrix.
Identity matrix ${displaystyle I_{n}}$ and the zero matrix are diagonal matrices.
Self-value ${displaystyle operatorname {diag} (a_{1},dotsa_{n})}$ They are. ${displaystyle a_{1},dotsa_{n}}$ .
Vectors ${displaystyle mathbf {e} _{1},dotsmathbf {e} _{n}}$ They form a self-voking base.

Uses

Diagonal matrices occur in many areas of linear algebra. Due to the simplicity of the operations on diagonal matrices and the computation of their determinant and their eigenvalues and eigenvectors, it is always desirable to represent a given matrix or linear transformation as a diagonal matrix.

In fact, a given n×n matrix is similar to a diagonal matrix if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.

In the field of real or complex numbers there are more properties: every normal matrix is similar to a diagonal matrix (see spectral theorem) and every matrix is equivalent to a diagonal matrix with non-negative entries.

Data: Q332791

Contenido relacionado

Más resultados...