H_
Nilpotent Transformations
SKIP_ p1 QM_[0;true;false;true;false;true;true;true;false] Let A and B be square n by n matrices.
  1. AS_[true;false] If A and B are similar matrices and A is nilpotent, then so is B.
  2. AS_[true;false] If A is nilpotent, then A has an inverse which is nilpotent.
  3. AS_[true;false] If A is nilpotent, then so is the transpose of A.
  4. AS_[true;false] If A has determinant zero, it is nilpotent.
  5. AS_[true;false] If A is nilpotent, then A has 0 as an eigenvalue.
  6. AS_[true;false] If A is nilpotent, then A has no eigenvalues except 0.
  7. AS_[true;false] If A is nilpotent, its index of nilpotency can be no larger than its dimension n.
  8. AS_[true;false] If A is nilpotent, its index of nilpotency can be no larger than its dimension n - 1.
SKIP_ p2 QM_[0;true;false;true;false;true;true] Let A be an n by n upper triangular matrix. AH_[2]
  1. AS_[true;false] Every entry on the diagonal of A is an eigenvalue of A.
  2. AS_[true;false] Every column vector with only one non-zero entry is an eigenvector of A.
  3. AS_[true;false] A is nilpotent if and only if all the diagonal entries of A are zero.
  4. AS_[true;false] If A is nilpotent, then images/ma322c5d1.png has the entry in the upper right hand corner zero, i.e. images/ma322c5d2.png .
  5. AS_[true;false] If A is nilpotent, then images/ma322c5d3.png has all its elements directly above the diagonal are zero. (i.e. images/ma322c5d4.png for i = 2, 3, ..., n.)
  6. AS_[true;false] If A is nilpotent and all the entries above the diagonal are positive, then the index of nilpotency is exactly n.
SKIP_ p3 QM_[0;n;true;r;r;1] Let A be an n by n matrix with entries on the superdiagonal non-zero (images/ma322c5d5.png for i = 2, 3, ..., n.) and with all other entries zero. AH_[2]
  1. The index of nilpotency of A is exactly AS_[1;2;n-1;n;n+1] .
  2. AS_[true;false] Let B be any nilpotent square n by n matrix and v be a non-zero vector. The vectors images/ma322c5d6.png for k = 0, 1, 2, ... span a subspace W of images/ma322c5d7.png and the linear map corresponding to B maps W into W.
  3. With B, v, and W as in the last part, let r be the smallest positive integer with images/ma322c5d8.png . Then the index of nilpotency of the linear map of W corresponding to B is exactly AS_[1;2;r-1;r;r+1;n-1;n;n+1] .
  4. With B, v, and W as in the last part, the dimension of W is precisely AS_[1;2;r-1;r;r+1;n-1;n;n+1] .
  5. With B, v, and W as in the last part, the matrix of the linear map from W to W corresponding to B is a matrix with subdiagonal elements all equal to AS_[0;1;2;r-1;r;r+1;n-1;n;n+1] (with respect to the basis of the images/ma322c5d9.png 's).
SKIP_ p4 QM_[0;3;2;3;1;0;4;5] Let T be a linear map which maps images/ma322c5d10.png and images/ma322c5d11.png where the images/ma322c5d12.png are a basis of images/ma322c5d13.png . AH_[2]
  1. The index of nilpotency of T is AS_[0;1;2;3;4;5] .
  2. The dimension of the nullspace of T is AS_[0;1;2;3;4;5] .
  3. The dimension of the image space TV is AS_[0;1;2;3;4;5] .
  4. The dimension of the space images/ma322c5d14.png is AS_[0;1;2;3;4;5] .
  5. The dimension of the space images/ma322c5d15.png is AS_[0;1;2;3;4;5] .
  6. The nullity of images/ma322c5d16.png is AS_[0;1;2;3;4;5] and the nullity of images/ma322c5d17.png is AS_[0;1;2;3;4;5] .
SKIP_ p5 QM_[0;4;5;3;1;0;2;4;6;7;e7;e5;e3;e1;e6;e4;e2] Let A be the 7 by 7 matrix with all entries 0 except that the diagonal above the superdiagonal is filled with entries equal to 1. Let V denote the vector space images/ma322c5d18.png . AH_[2]
  1. The matrix A has index of nilpotency AS_[0;1;2;3;4;5;6;7] .
  2. The dimensions of images/ma322c5d19.png , images/ma322c5d20.png , images/ma322c5d21.png , and images/ma322c5d22.png are AS_[0;1;2;3;4;5;6;7] , AS_[0;1;2;3;4;5;6;7] , AS_[0;1;2;3;4;5;6;7] , and AS_[0;1;2;3;4;5;6;7] respectively.
  3. The nullities of images/ma322c5d23.png , images/ma322c5d24.png , images/ma322c5d25.png , and images/ma322c5d26.png are AS_[0;1;2;3;4;5;6;7] , AS_[0;1;2;3;4;5;6;7] , AS_[0;1;2;3;4;5;6;7] , and AS_[0;1;2;3;4;5;6;7] respectively.
  4. If e1, e2, ..., e7 are the standard basis of images/ma322c5d27.png , then the linear map represented by A can be expressed as AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d28.png AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d29.png AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d30.png AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d31.png and AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d32.png AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d33.png AS_[e1;e2;e3;e4;e5;e6;e7] images/ma322c5d34.png .
SKIP_ p6 QM_[0;2;0;1;0;0] The A be the matrix

images/ma322c5d35.png

AH_[2] When put in canonical form, the matrix is seen to have AS_[0;1;2;3;4;5;6;7] strings with 1 basis element, AS_[0;1;2;3;4;5;6;7] strings with 2 basis elements, AS_[0;1;2;3;4;5;6;7] strings with 3 basis elements, AS_[0;1;2;3;4;5;6;7] strings with 4 basis elements, and AS_[0;1;2;3;4;5;6;7] strings with 5 basis elements. SKIP_ p7 QM_[0;true;true] Let A be a square 2 by 2 nilpotent matrix. AH_[2]

  1. AS_[true;false] If the index of nilpotency is 1, then A is the zero matrix.
  2. AS_[true;false] If the index of nilpotency is 2, then A has a string with two basis elements (i.e. one of the form images/ma322c5d36.png ).
SKIP_ p8 QM_[0;true;2;3] Let A be a square 3 by 3 nilpotent matrix. AH_[2]
  1. AS_[true;false] If the index of nilpotency is 1, then A is the zero matrix.
  2. If the index of nilpotency is 2, then A can be represented with a basis in which there is a string with one basis element and another string with AS_[0;1;2;3] basis elements.
  3. If the index of nilpotency is 3, then A can be respresented with a basis in which there is a single string with AS_{0;1;2;3] basis elements.
SKIP_ p9 QM_[0;true;false;false;true] Let A be a square 4 by 4 nilpotent matrix. AH_[2]
  1. AS_[true;false] If the index of nilpotency is 1, then A is the zero matrix.
  2. AS_[true;false] If the index of nilpotency is 2, then A can be represented with a basis in which there are two strings with two basis elements each.
  3. AS_[true;false] If the index of nilpotency is 3, then A can be represented with a basis in which there are two strings with one basis element and another with two basis elements.
  4. AS_[true;false] If the index of nilpotency is 4, then A can be represented with a basis in which there is a single string with 4 basis elements.
SKIP_ p10 QM_[0;true;true;true;true] Let A be a square matrix. The characteristic polynomial of A is the polynomial p(X) = det(A - Ix) where I is the n by n identity matrix.
  1. AS_[true;false] If B is similar to A, then A and B have the same characteristic polynomials.
  2. AS_[true;false] If B is similar to A and p, q are the characteristic polynomials of A and B respectively, then p(A) = 0 if and only if q(B) = 0.
  3. AS_[true;false] If A is nilpotent and p is the characteristic polynomial of A, then p(A) = 0.
  4. AS_[true;false] If A is upper triangular with all its diagonal elements equal, then p(A) = 0 where p is the characteristic polynomial of A.
SKIP_ p11 QM_[0;true] AH_[2] AS_[true;false] If A is diagonalizable, then p(A) = 0 where p is the characteristic polynomial of A. SKIP_