Section 3 Summary

Home , Identity matrix, Invertible matrix, Permutation matrix

Section 3 summary

Brian Krummel March 16, 2020

1 Terminology

Transformation

Domain

Codomain or target space

Image under a transformation

Matrix transformation

Linear transformation

Standard matrix of a linear transformation

Matrix multiplication

Row-column rule

Partitioned matrix

Invertible matrix

Nonsingular matrix

Singular matrix

Inverse matrix

Invertible transformation

Inverse transformation

LU-factorization

PLU-factorization

Type-2 matrix

Unipotent matrix

1 Permutation matrix Coordinate vector Point matrix Coordinate matrix Point transformation Coordinate transformation Linear isomorphism Isomorphic Matrix of a linear transformation with respect to bases on the domain and codomain

2 Theorems about linear transformations

n m Theorem 1. Let TA : R → R be the matrix transformation TA(X) = AX given by multiplication by an m × n matrix A. Then the image of a line segment under TA is either a point or a line segment. Theorem 2. Let T : V → W be any linear transformation. • T (0) = 0).

• T (c1X1 + c2X2 + ··· + ckXk) = c1 T (X1) + c2 T (X2) + ··· + ck T (Xk) for all vectors Xi ∈ V and all scalars ci ∈ R. Theorem 3 (Matrix Representation Theorem). Let T : Rn → Rm be a linear transformation. There exists a unique m × n matrix A such that T (X) = AX for all X ∈ Rn. In particular, A = T (E1) T (E2) ··· T (En) n where Ej ∈ R is the coordinate vector with j-th entry 1 and all other entries zero. (Equivalently, Ej is the j-th column of the n × n identity matrix I.) n m Theorem 4. Let TA : R → R be the matrix transformation TA(Y ) = AY given by multiplication by an m × n matrix A. Let B be a basis for Rn and D be a basis for Rm. Then there exists a unique m × n matrix M of TA with respect to the bases B and D such that

CDTA(Y ) = CDAY = MCBY for all Y ∈ Rn; that is, the D-coordinates of AY are equal to M times the B-coordinate vector X of Y . In particular, −1 M = PD APB or equivalently −1 A = PDMPB .

2 Theorem 5. Let L : V → W be a linear transformation. Let B = {B1,B2,...,Bn} be a basis for V and D be a basis for W. Then there exists a unique matrix M of T with respect to the bases B and D such that C C TD (L(Y )) = MTB (Y ) for all Y ∈ V; that is, the D-coordinates of L(Y ) are equal to M times the B-coordinate vector X of Y . In particular,

C C C M = TD (L(B1)) TD (L(B2)) ··· TD (L(Bn)) so that the j-th column of M is the D-coordinates of L(Bj) and the (i, j)-entry of M is the Di- coeﬃcient of L(Bj).

3 Properties of matrix multiplication

Theorem 6. Let A, B, C be matrices of appropriate sizes and I denote the identity matrix of appropriate sizes.

• A(BC) = (AB)C

• A(B + C) = AB + AC

• (A + B)C = AC + BC

• r(AB) = (rA)B = A(rB) for every scalar r ∈ R • IA = AI = A

• (AB)T = BT AT

Warnings! Let A, B, C be matrices of appropriate sizes. In general,

• AB 6= BA

• AB = AC does not imply B = C

• AC = BC does not imply A = B

• AB = 0 does not imply B = 0 or C = 0.

Theorem 7. Let A, B be matrices of appropriate sizes. Then rank(AB) ≤ rank A and rank(AB) ≤ rank B.

4 Invertible matrices

Deﬁnition 1. A is an invertible n × n matrix if there exists a matrix A−1 such that

AA−1 = I andA−1A = I.

Whenever A−1 exists, it is also unique. A singular n×n matrix is a matrix which is not invertible.

3 Algorithm for computing A−1. Apply Gaussian elimination A I −→ I A−1 .

Theorem 8. If A is an invertible n × n matrix, then for every B ∈ Rn the equation AX = B has exactly one solution X = A−1B.

Theorem 9. Consider the matrix transformation TA(X) = AX where A is an n×n matrix. TA is −1 −1 an invertible transformation if and only if A is an invertible matrix, in which case TA (X) = A X for all X ∈ Rn. Theorem 10.

• If A is an invertible n×n matrix, then A−1 is also an invertible n×n matrix and (A−1)−1 = A.

• If A and B are invertible n × n matrices, then AB is also an invertible n × n matrix and (AB)−1 = B−1A−1.

• If A is an invertible n × n matrix, then AT is also an invertible n × n matrix and (AT )−1 = (A−1)T .

Theorem 11 (Invertible Matrix Theorem). Let A be an n × n square matrix. The following are logically equivalent: o (a) A is an invertible matrix. Main statement  (b) rank A = n.   (c) A has n pivot positions. Echelon form  (d) A is row equivalent to the n × n identity matrix.   (e) The homogeneous equation AX = 0 has only the trivial solution.   (f) The columns of A form a linearly independent set. Uniqueness  (g) Nul A = {0}.   n (h) The equation AX = B has at least one solution for each B ∈ R .   (i) The columns of A span Rn. Existence  (j) Col A = Rn.   n (k) The equation AX = B has exactly one solution for each B ∈ R .   (l) The columns of A are a basis for Rn. Exists and unique   (m) The matrix transformation TA(X) = AX is invertible.   (n) There is an n × n matrix C such that CA = I.  Deﬁnition of inverse (o) There is an n × n matrix D such that AD = I.   (p) AT is invertible.  Transpose (q) Row A = Rn. 

4 5 LU-Factorization

Theorem 12. Let A be a type-2 m × n matrix. Suppose that A I is row equivalent to U V for some m × n matrix U and m × m matrix V :

A I −→ U V .

Then V is invertible and A = V −1U.

Theorem 13. Let A be a type-2 m × n matrix. Then there exists an m × m lower trangular, unipotent matrix L (i.e. L is lower triangular and has entries one on the main diagonal) and there exists an m × n matrix U in echelon form such that

A = LU.

Algorithm for computing an LU-factorization. Assuming A is a type-2 n × n matrix, apply Gaussian elimination A I −→ U V . At the same time, write down a lower-triangular n × n matrix L with diagonal entries 1. For each row replacement Ri − c Rj 7→ Ri, which changes the (i, j)-entry of A to a zero, set the (i, j)-entry of L equal to +c (with the opposite sign). To solve AX = B, we can ﬁnd the LU-factorization A = LU (if it exists). Then we use back-substitution to solve for Z such that LZ = B. Then we again back-substitution to solve for X such that UX = Z.

Theorem 14. Let A be any non-type-2 m × n matrix. Then there exists an m × n matrix A0 with the same rows as A in a diﬀerent order such that A0 is a type-2 matrix.

Theorem 15. Let A be any m × n matrix. Then there exists an m × m permutation matrix P , m × m lower trangular, unipotent matrix L, and there exists an m × n matrix U in echelon form such that A = P LU.