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Diagonal matrix
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Diagonal matrix
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. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, . In geometry, a diagonal matrix may be used as a scaling matrix, since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale.
![{\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd1681ed0048aa1b28a2e9b5f6bb834b1cc3b14)
![{\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&5&0\\0&0&4\end{smallmatrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bad34bc0da45cc29d4095720a4a8c9b2cac2a26)
![{\displaystyle \left[{\begin{smallmatrix}0.5&0\\0&0.5\end{smallmatrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae9d50cf77850fc91f92a367564eafc7f1f4718)
As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix D = (di,j) with n columns and n rows is diagonal if
However, the main diagonal entries are unrestricted.
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with all the entries not of the form di,i being zero. For example:
More often, however, diagonal matrix refers to square matrices, which can be specified explicitly as a square diagonal matrix. A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix.
The following matrix is square diagonal matrix:
If the entries are real numbers or complex numbers, then it is a normal matrix as well.
In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".
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Diagonal matrix
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. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it is a diagonal matrix called a scalar matrix, for example, . In geometry, a diagonal matrix may be used as a scaling matrix, since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale.
As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix D = (di,j) with n columns and n rows is diagonal if
However, the main diagonal entries are unrestricted.
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with all the entries not of the form di,i being zero. For example:
More often, however, diagonal matrix refers to square matrices, which can be specified explicitly as a square diagonal matrix. A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix.
The following matrix is square diagonal matrix:
If the entries are real numbers or complex numbers, then it is a normal matrix as well.
In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".