Iyad's Page of Matrices

A square matrix A=(a_ij) of order n is said to be:

1. Skew-symmetric if A^t=-A.
   
2. Symmetric if A^t=A.
   
3. Hermitian if A^*=A.
   
4. Skew-Hermitian if A^*=-A.
   
5. Diagonal if all nondiagonal elements are equal to zero.
   
6. Of rank-one if it has one linearly independent row only.
   
7. Idempotent if A²=A.
   
8. Involutory if A²=I, where I is the identity matrix.
   
9. Orthogonal if A^-1=A^t.
   
10. Unitary if A^-1=A^*.
    
11. Upper-triangular if all elements below the main diagonal are equal to zero. I.e, a_ij=0 for j < i.
    
12. Lower-triangular if all elements above the main diagonal are equal to zero. I.e, a_ij=0 for i < j.
    
13. Upper Hessenberg if a_ij=0 for i > j+1.
    
14. Lower Hessenberg if a_ij=0 for j > i+1.
    
15. Positive definite if x^* A x > 0 for every vector x.
    
16. Positive semi-definite if x^* A x >= 0 for every vector x.
    
17. Negative definite if x^* A x < 0 for every vector x.
    
18. Negative semi-definite if x^* A x <= 0 for every vector x.
    
19. Nilpotent if there exists a positive integer k such that A^k = 0.
    
20. Of full rank if it has n linearly independent rows.
    
21. Defective if it has less than n linearly independent eigenvectors.
    
22. Ill-conditioned if Cond(A) is quite large, where .
    
23. Singular if det (A)=0.
    
24. Invertible if it is nonsingular.
    
25. Normal if AA^*=A^*A.
    
26. Toeplitz if the elements along each diagonal are the same.
    
27. Identity if a_{i i}=1 for all i and all other elements are equal to zero.
    
28. Counteridentity if a_{i n-i+1}=1 for all i and all other elements are equal to zero. In other words, all elements on the main counterdiagonal are equal to 1 and the rest are equal to zero. The counteridentity matrix is denoted by J.
    
29. Centrosymmetric if it is symmetric about the center. In other words, JAJ=A, where J is the counteridentity matrix. I.e. A(i,j)=A(n+1-i,n+1-j), where n is the order of the matrix.
    
30. Skew-centrosymmetric if it is skew-symmetric about the center. In other words, JAJ=-A. I.e. A(i,j)=-A(n+1-i,n+1-j), where n is the order of the matrix.
    
31. Persymmetric if it is symmetric about the main counterdiagonal. In other words, JAJ=A^t.
    
32. Permutation if every row and column has exactly one nonzero element which is 1.
    
33. Housholder if it is of the form I - 2 v v^t/(v^tv) for some nonzero n-vector v.
    
34. Stable if the real part of each one of its eigenvalues is negative.

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