Chapter 2: Matrix Algebra Flashcards
Matrix
a rectangular array of numbers called
the entries or elements of the matrix.
For a matrix A we write the entry in row i, column j as …
a_ij
if A is an m × n matrix then A has …
m rows and n columns
Square matrix
if m=n
diagonal matrix
iff its off-diagonal elements are zero (for i 6= j, aij = 0)
The n × n identity matrix In is the diagonal
matrix with all diagonal entries equal to …
1
Two matrices are equal if and only if
they have the same number of rows and the same number of columns and the corresponding entries are equal.
If A = (aij) and B = (bij) are m × n matrices, then we define the new m × n matrix A + B componentwise:
A + B = (aij) + (bij) = (aij + bij);
If A = (aij) is an m × n matrix and c is a scalar then we define the new m × n matrix cA componentwise:
cA = c(aij) = (caij).
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(a) A + B =
B + A
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(b) (A + B) + C =
A + (B + C).
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(c) There is an m × n matrix 0 such that
A + 0 = A, for each A
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(d) For each A there is an m × n matrix, −A such that
A + (−A) = 0.
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(e) c(A + B) =
cA + cB
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(f) (c + d)A =
cA + dA
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(g) c(dA) =
(cd)A.
Theorem 2.1.1 Let A, B and C be m × n matrices and c, d ∈ R.
Then
(h) 1A =
A
Transpose of an m × n matrix A. written A^T
the n × m matrix whose rows are the columns of A in the same order.
If A = [aij] then A^T =
[aji].
Theorem 2.1.2 Let A and B be matrices of the same size, and let k ∈ R
(a) (A^T)^T =
A
Theorem 2.1.2 Let A and B be matrices of the same size, and let k ∈ R
(b) (A + B)^T =
A^T + B^T
Theorem 2.1.2 Let A and B be matrices of the same size, and let k ∈ R
(c) (kA)^T =
k(A^T)
Symmetric square matrix
if A^T = A.
(That is, A is equal to its own transpose)
If A and B are square symmetric matrices, then A + B is
symmetric
If A is a square matrix (not necessarily symmetric), then A + A^T is
a symmetric matrix.
Ordered n-tuple
An ordered sequence (a1, a2, . . . , an) of real numbers
Definition. Let R denote the set of all real numbers. The set of all ordered n-tuples from R is denoted by
R^n.
R^n = {(x1, x2, · · · , xn) : x1, · · · , xn ∈ R} .
If x, y ∈ R^n it is clear that their matrix sum x + y is also in R^n as is the scalar multiple kx for any real number k. We express this observation by saying that
R^n is closed under addition and scalar
multiplication.
Definition A vector v is a linear combination of vectors v1, · · · , vk if and only if
there are scalars c1, c2, · · · , ck such that
v = c1v1 + c2v2 + · · · + ckvk.
Theorem 2.2.1
a) Every system of linear equations has the form
Ax = b where A is the coefficient matrix, b is the constant matrix, and x is the matrix of variables.
Coefficients of the linear combination
The scalars c1, c2, · · · , ck in v = c1v1 + c2v2 + · · · + ckvk.
Theorem 2.2.1
b) The system Ax = b is consistent if and only if
b is a linear combination of the columns of A
Theorem 2.2.1
c) If a1, a2 . . . , an are the columns of A and if x = [x1,…xm] (as a column vector), then
x is a solution to the linear system Ax = b if and only if
x1, x2, . . . , xn are a solution of the vector equation x1a1 + x2a2 + · · · + xnan = b
Definition. A function or mapping or transformation T from R^m to R^n is a
rule that assigns to each vector x ∈ R^m a unique vector T(x) ∈ R^n
The domain of T in the transformation T from R^m to R^n is
R^m
The codomain of T in the transformation T from R^m to R^n is
R^n
To indicate that T is a map with domain R^m and codomain R^n, write
T: R^m –> R^n
If x is in the domain of T, the image of x is
the vector T(x)
The range of T in the transformation T from R^m to R^n is
the set of all possible images,
i.e. range(T) = {T(x) : x ∈ R^m}.
Definition. If A is any m × n matrix, multiplication by A gives
a matrix transformation T_A : R^n → R^m defined by T_A(x) = Ax for every x in Rn
Relection in x-axis is achived by multiplying by the matrix
l1 0 l (Check this with vector [x,y]
L0 -1J T([x,y])=[x,-y]
Counterclockwise rotation about the origin through pi/2 radians
l0 -1l T([x,y])=[-y,x]
L1 0J
x-expansion (compression)
la 0l T([x,y])=[ax,y]
L0 1J
For a > 1 we have expansion and for a < 1 we have compression
x-shear
l1 al T([x,y])=[x+ay,y]
L0 1J
For a > 0 we have a positive x-shear and for a < 0 we have a negative x-shear
The zero transformation
If A is the m × n zero matrix, T_0 : R^n → R^m is given by T_0(v) = 0R^m for all v ∈ R^n
The identity transformation
If A is the n × n identity matrix, I_n : R^n → R^n given by I_n(v) = v for all v ∈ Rn
Let T : R^k → R^m and S : R^n → R^m be transformations. The composition S ◦ T is the mapping S ◦ T : R^k → R^m defined by
(S ◦ T)(x) =
S(T(x))
for all x ∈ R^n.
Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(f) (AB)^T =
B^TA^T
Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(e) k(AB) =
(kA)B = A(kB)
Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(d) (A + B)C =
AC + BC
Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(c) A(B + C) =
AB + AC
Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(b) A(BC) =
(AB)C
Theorem 2.3.3 Let A, B and C be matrices and k be a scalar. The following identities hold whenever the matrix products are defined:
(a) I_m A =
A = AI_n if A is m × n
Inverse of A (n × n matrix)
an n × n matrix A’ such that AA’ = I_n, and A’A = I_n.
A is invertible
if A’ exists
A is singular or not invertible
If no inverse exists
Theorem Uniqueness of the matrix inverse. If an n × n matrix A is invertible then
its inverse is unique.
Theorem 2.4.2 If A is an invertible n × n matrix then the system of linear equations given by Ax = b has the unique solution given by
x = A^−1b.
If A = [a b]
[c d]
then A is invertible if ______
in which case A^-1= _______
- ad-bc =/= 0
- 1/(ad-bc)*[d -b]
[-c a]
Gauss–Jordan method for computing the inverse.
If A is an invertible matrix, there exists a sequence of elementary row operations that carry A to the identity matrix I of the same size, written A → I. This same series of row operations carries I to A^−1. The algorithm can be summarised as follows:
[A | I] –ERO’s–> [I | A^-1]
Theorem 2.4.4
(a) If A is an invertible matrix, then A^−1 is _____ and (A^−1)^−1 = ____
- invertible
- A.
Theorem 2.4.4
(b) If A is an invertible matrix and c =/= 0 is a scalar then cA is ____ and (cA)^−1 = ______
- invertible
- 1/c*A^−1
Theorem 2.4.4
(c) If A and B are invertible matrices of the same size, then AB is _____ and (AB)^−1 = _____
- invertible
- B^−1A^−1
Theorem 2.4.4
(d) If A is an invertible matrix, then A^T
is _____ and (A^T)^−1 = ______
- invertible
- (A^−1)^T
Theorem 2.4.4
(e) If A is invertible matrix then A^n is ____ for all integers n ≥ 0 and (A^n)^−1 = ______
- invertible
- (A^−1)^n
Definition If A is invertible and n ≥ 0 an integer we define A^−n = ____ = ____
(A^−1)^n = (A^n)^−1
Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is _______;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.
invertible
Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution _____;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.
x = 0
Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is ____;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.
I_n
Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has __ ____ _____ for every b in R^n
(e) There exists an n × n matrix C such that AC = I_n.
a unique solution
Theorem 2.4.5 The fundamental theorem of invertible matrices (Version 1) Let A be an n × n matrix. The following statements are equivalent:
(a) A is invertible;
(b) Ax = 0 has only the trivial solution x = 0;
(c) The reduced echelon form of A is I_n;
(d) Ax = b has a unique solution for every b in R^n
(e) There exists an n × n matrix C such that AC = ____
I_n.
A matrix transformation is invertible
if there exists a matrix transformation T’: Rn → Rn with T’ ◦ T = I_n and T ◦ T’ = I_n
If the matrix transformation T has an inverse, then its matrix A must be
invertible
The geometric view of the inverse of a transformation provides a way to find the inverse of the matrix A. For example if the matrix A induces a reflection about the x-axis in R2, then A^−1 can be found by …
finding the transformation that reverses that reflection
Elementary matrix
an n × n matrix which can be obtained by performing one elementary row operation on
an identity matrix In. We say that E is of type I, II or III if the operation is of that type.
Performing elementary row operations is entirely equivalent to
multiplying by elementary matrices
Lemma 2.5.1 Let E be the elementary matrix obtained by performing an elementary row operation on Im. If the same elementary row operation is performed on an m × n matrix A, then the result is the matrix ___
EA.
Lemma 2.5.2 Each elementary matrix is _____, and its inverse is _______.
- invertible
- an elementary matrix of the same type
Theorem 2.5.1 Suppose A is an m × n matrix and A → B by
elementary row operations. Then,
a) B =______
b) U can be computed by [A | Im] –ERO’s–> [B | U] using the operations carrying A → B.
c) U = E_kE__k−1. . . E_2E_1 where E_1, E_2, . . . , E_k are the elementary matrices corresponding (in order) to the elementary row operations carrying A to B.
UA where U is an m × m invertible matrix .
Theorem 2.5.1 Suppose A is an m × n matrix and A → B by
elementary row operations. Then,
a) B = UA where U is an m × m invertible matrix .
b) U can be computed by ______
c) U = E_kE__k−1. . . E_2E_1 where E_1, E_2, . . . , E_k are the elementary matrices corresponding (in order) to the elementary row operations carrying A to B.
[A | Im] –ERO’s–> [B | U] using the operations carrying A → B.
Theorem 2.5.1 Suppose A is an m × n matrix and A → B by
elementary row operations. Then,
a) B = UA where U is an m × m invertible matrix .
b) U can be computed by [A | Im] –ERO’s–> [B | U] using the operations carrying A → B.
c) U = _____
E_kE__k−1. . . E_2E_1 where E_1, E_2, . . . , E_k are the elementary matrices corresponding (in order) to the elementary row operations carrying A to B.
Theorem 2.5.2 A square matrix is invertible if and only if
it is a product of elementary matrices.
Definition A linear transformationfrom Rn
to Rm is a mapping T : Rn → Rm such that for all x, y ∈ Rn and scalars c ∈ R,
a) _____;
b) T(cx) = cT(x).
T(x + y) = T(x) + T(y)
Definition A linear transformationfrom Rn
to Rm is a mapping T : Rn → Rm such that for all x, y ∈ Rn and scalars c ∈ R,
a) T(x + y) = T(x) + T(y);
b) _____.
T(cx) = cT(x)
Theorem 2.6.2 Let T : Rn → Rm be a linear transformation. Then T is a ______ induced by the unique m × n matrix A, where the jth column of A is T(ej and A =[ T(e1) T(e2) · · · T(en) ], where {e1, e2, . . . , en} is _____, that is the columns of the identity matrix I_n
- matrix transformation
- the standard basis for Rn
