Numerical linear algebra comes up with different types of matrices. While reading the documentation for linear algebra libraries (such as LAPACK), you may inadvertently forget the matrix definitions written there. Also, although I somehow remember the definition, I often want to imagine a concrete matrix shape. From time to time, you may also want to know how to read English. It takes a surprising amount of time to look up them in reference books and on the Internet. ** **
** Therefore, in this article, we have listed the definitions and concrete examples of matrices that frequently appear in linear algebra by calculation. ** ** I hope you can save the above-mentioned trouble as much as possible.
If you point out any deficiencies or deficiencies due to my lack of study, I will revise it, so please contact me in that case.
A matrix with most of its components zero.
M=
\begin{pmatrix}
1 & 2 & 0 & 0\\
0 & 4 & 0 & 0\\
0 & 0 & 7 &0\\
0 & 3 & 0 &0\\
\end{pmatrix}
A matrix with few zero element components.
M=
\begin{pmatrix}
1 & 4 & 6 & 0\\
0 & 4 & 3 & 2\\
5 & 1 & 7 &3\\
5 & 3 & 6 &1\\
\end{pmatrix}
A square matrix with $ a_ {ij} = 0 \ (i = 2, ..., n; j = 1, .., i-1) $ with all other non-zero elements.
M=
\begin{pmatrix}
1 & 2 & 5 & 5\\
0 & 4 & 4 & 2\\
0 & 0 & 7 &11 \\
0 & 0 & 0 &1\\
\end{pmatrix}
A square matrix with $ a_ {ij} = 0 \ (i = 2, ..., n-1; j = i + 1, ..., n) $ with all other non-zero elements.
M=
\begin{pmatrix}
1 & 0 & 0 & 0\\
3 & 4 & 0 & 0\\
5 & 5 & 2 &0 \\
6 & 2 & 9 &10\\
\end{pmatrix}
Only the diagonal component of the square matrix is non-zero and the off-diagonal component is zero.
M=
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 4 & 0 & 0\\
0 & 0 & 6 &0 \\
0 & 0 & 0 &10\\
\end{pmatrix}
Non-zero components are concentrated near the diagonal.
M=
\begin{pmatrix}
1 & 2 & 0 & 0 &0 \\
2 & 4 & 5 & 0 &0\\
0 & 3 & 6 &1 &0\\
0 & 0 & 2 &5 &4\\
0 & 0 & 0 &1 &9\\
\end{pmatrix}
In a square matrix, all diagonal elements and diagonal elements adjacent to them are not zero, and all other elements are zero.
M=
\begin{pmatrix}
1 & 2 & 0 & 0\\
3 & 4 & 8 & 0\\
0 & 5 & 6 &9\\
0 & 0 & 1 &10\\
\end{pmatrix}
In a square matrix, $ a_ {ij} = 0 \ (i = 3,4, ..., n; j = 1, 2, ..., i-2) $, and the other components are non-zero.
M=
\begin{pmatrix}
1 & 2 & 8 & 20 & 6\\
3 & 4 & 8 & 2 & 4\\
0 & 5 & 6 &9 & 1 \\
0 & 0 & 6 &2 & 5\\
0 & 0 & 0 &3 & 10\\
\end{pmatrix}
In a square matrix $ a_ {ij} = 0 \ (i = 1,2, ..., n-2; j = i + 2, i + 3, ..., n) $, other components are non-zero Zero thing.
M=
\begin{pmatrix}
1 & 2 & 0 & 0 & 0\\
3 & 4 & 8 & 0 & 0\\
2 & 5 & 6 &4 & 0 \\
7 & 4 & 6 &2 & 5\\
9 & 5 & 3 &3 & 10\\
\end{pmatrix}
The expansion of the characteristic polynomial (characteristic equation) $ det (A -tI) $ is when $ A $ is a $ nxn $ matrix.
Will be. Now consider n = 5 as an example.
At this time, the ** companion matrix of matrix A ** has the following form.
C =
\begin{pmatrix}
0 & 0 & 0 & 0 & -c_5\\
1 & 0 & 0 & 0 & -c_4\\\
0 & 1 & 0 & 0 & -c_3 \\
0 & 0 & 1 & 0 &-c_2\\
0 & 0 & 0 & 1 & -c_1\\
\end{pmatrix}
A square matrix whose diagonal component is 1 and the others are zero. Represented as the symbols $ I $ and $ E $.
I =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0& 0 \\
0 & 0 & 0 &1 & 0\\
0 & 0 & 0 &0 & 1\\
\end{pmatrix}
A matrix in which all genders are zero. Represented by the symbol $ O $.
O =
\begin{pmatrix}
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0& 0 \\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 &0 & 0\\
\end{pmatrix}
The relationship of $ a_ {ij} = a_ {ji} $.
M=
\begin{pmatrix}
1 & 2 & 9 & 3\\
2 & 4 & 2 & 0\\
9 & 2 & 7 &3\\
3 & 0 & 3 & 10\\
\end{pmatrix}
A square matrix with a diagonal component of zero and an off-diagonal component of $ a_ {ij} =-a_ {ji} $.
M=
\begin{pmatrix}
0 & 2 & 9 & -3\\
-2 & 0 & 2 & 0\\
-9 & -2 & 0 &-3\\
3 & 0 & 3 & 0\\
\end{pmatrix}
In the transposed complex conjugate $ H ^ \ dagger $ of the square matrix $ H $,
A matrix that satisfies $ H ^ \ dagger = H $.
H=
\begin{pmatrix}
5 & 2+i & 9 & 3+i\\
2-i & 0 & -2-4i & 7-6i\\
9 & -2+4i & 4 &3\\
3-i & 7+6i & 3 & 4i\\
\end{pmatrix}
Between the square matrix $ U $ and its transposed complex conjugate $ U ^ \ dagger $,
A matrix with the relationship. $ I $ is the identity matrix.
M=
\begin{pmatrix}
1 & i \\
-i & 2 \\
\end{pmatrix}
The transposed matrix $ B $ of the matrix $ A $ has the relation of $ b_ {ij} = a_ {ji} $.
Put $ B $ in $ A ^ T $.
Between the square matrix M and its transposed matrix $ M ^ T $,
The relationship of $ M M ^ T = M ^ T M = I $ holds.
M=
\begin{pmatrix}
0 & 1 \\
1 & 0 \\
\end{pmatrix}
The following books have been helpful in writing this article.
[1] Gilbert Strang, ["World Standard MIT Textbook Strang: Linear Algebra Introduction"](https://www.amazon.co.jp/%E4%B8%96%E7%95%8C%E6%A8%99 % E6% BA% 96MIT% E6% 95% 99% E7% A7% 91% E6% 9B% B8-% E3% 82% B9% E3% 83% 88% E3% 83% A9% E3% 83% B3% E3% 82% B0-% E7% B7% 9A% E5% BD% A2% E4% BB% A3% E6% 95% B0% E3% 82% A4% E3% 83% B3% E3% 83% 88% E3 % 83% AD% E3% 83% 80% E3% 82% AF% E3% 82% B7% E3% 83% A7% E3% 83% B3-% E3% 82% AE% E3% 83% AB% E3% 83% 90% E3% 83% BC% E3% 83% 88 / dp / 4764904055 / ref = pd_lpo_sbs_14_t_0? _ Encoding = UTF8 & psc = 1 & refRID = 9817PCQXDR5497M5GPS2), Modern Science, 2015.