- 3.2.1: Let A be an m x n matrix. Prove that there exists a sequence of ele...
- 3.2.2: Label the following statements as true or false. (a) The rank of a ...
- 3.2.3: Find the rank of the following matrices. (a) (c) 1 0 2 1 1 4 166 Ch...
- 3.2.4: Prove that for any mxn matrix A, rank(A) 0 if and only if A is the ...
- 3.2.5: Use elementary row and column operations to transform each of the f...
- 3.2.6: For each of the following matrices, compute the rank and the invers...
- 3.2.7: Express the invertible matrix (! ) as a product of elementary matri...
- 3.2.8: Let A be an m x n matrix. Prove that if c is any nonzero scalar, th...
- 3.2.9: Complete the proof of the corollary to Theorem 3.4 by showing that ...
- 3.2.10: Prove Theorem 3.6 for the case that A is an m x 1 matrix.
- 3.2.11: Let B = where B' is an m x n submatrix of B. Prove that if rank(B) ...
- 3.2.12: Let B' and D'bemx n matrices, and let B and D be (ra-f- 1) x (n + 1...
- 3.2.13: Prove (b) and (c) of Corollary 2 to Theorem 3.6.
- 3.2.14: Let T, U: V > W be linear transformations. (a) Prove that R(T + U) ...
- 3.2.15: Suppose that A and B are matrices having n rows. M(A\B) = (MA\MB) f...
- 3.2.16: Supply the details to the proof of (b) of Theorem 3.4.
- 3.2.17: Prove that if B is a 3 x 1 matrix and C is a 1 x 3 matrix, then the...
- 3.2.18: Let A be an m x n matrix and B be an n x p matrix. Prove that AB ca...
- 3.2.19: Let A be an m x n matrix with rank m and B be an n x p matrix with ...
- 3.2.20: Let / A = (a) Find a 5 x 5 matrix M with rank 2 such that AM = O, w...
- 3.2.21: Let A be an m x n matrix with rank m. Prove that there exists an nx...
- 3.2.22: Let B be an n x m matrix with rank m. Prove that there exists an mx...
Solutions for Chapter 3.2: The Rank of a Matrix and Matrix Inverses
Full solutions for Linear Algebra | 4th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).