Define diagnol matrix with example
Answers
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-by-2 diagonal matrix is , while an example of a 3-by-3 diagonal matrix is. .
Properties
The determinant of diag(a1, ..., an) is the product a1...an.
The adjugate of a diagonal matrix is again diagonal.
A square matrix is diagonal if and only if it is triangular and normal.
Any square diagonal matrix is also a symmetric matrix.
A symmetric diagonal matrix can be defined as a matrix that is both upper- and lower-triangular. The identity matrix In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.
Applications
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or linear map by a diagonal matrix.
In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that X−1AX is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.
Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA∗ = A∗A then there exists a unitary matrix U such that UAU∗ is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that UAV∗ is diagonal with positive entries.
Operator theory
In operator theory, particularly the study of PDEs, operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a separable partial differential equation. Therefore, a key technique to understanding operators is a change of coordinates—in the language of operators, an integral transform—which changes the basis to an eigenbasis of eigenfunctions: which makes the equation separable. An important example of this is the Fourier transform, which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the heat equation.
Especially easy are multiplication operators, which are defined as multiplication by (the values of) a fixed function–the values of the function at each point correspond to the diagonal entries of a matrix.
Step-by-step explanation: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-by-2 diagonal matrix is {\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}{\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}, while an example of a 3-by-3 diagonal matrix is{\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&7&0\\0&0&4\end{smallmatrix}}\right]}{\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
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