6.4 Symmetric Matrices
Definition: symmetric matrix
A is symmetric if A T A.
Eigen Properties of a Symmetric Matrix:
Let A be symmetric.
(1) Eigenvalues of A are real.
(2) Eigenvectors of A are linearly independent.
(3) Eigenvectors can be chosen so that
3.1 Spaces of Vectors
Definition: Vector Spaces:
Let V be a set of vectors. Let u + v be a vector addition and cu be a scalar multiplication for vectors u and v in
V and c a scalar. V with this vector addition and scalar multiplication is if eight rules h
7.1 Linear Transformations
Definition: transformation
T is a transformation from V to W if T assigns to each vector in V a unique vector in W.
Examples:
(1) Tv 1 , v 2 v 1 v 2
(2) Tv 1 , v 2 v 1 , v 2 1, 1
(3) Tv 1 , v 2 1 v 1 v 2 , 1 v 1 v 2
2
2
Definit
Math 530 - Fall, 2012
2
1. (18pts) Let
v
Exam 1
,w
3
Name: _Solutions_
2
1
, and
z
1
2
3
a. Compute w and determine if and w are parallel, perpendicular or neither.
v
v
w 2 3 1 0 so and w are not perpendicular.
v
v
cw so and w are not parallel.
v
6.2 Diagonalizing a Square Matrix
Definition: similar matrix
Let A and B be square matrices of the same size. A and B are similar if there exists an invertible matrix
S such that B SAS 1 .
Properties:
(1) If A and B are similar, then A and B have the same
6.1 Eigenvalues and Eigenvectors of a Square Matrix
Let A be an m n matrix. The function Tx Ax is also called a linear transformation. T transforms
a vector x in R n to a vector Ax in R m . When m n, both x and Ax are in R n .
Examples:
1 1
1 1 2
3 1
cos
2.7 Transpose and Permutation Matrices
Let A be an m n matrix.
Definition: (transpose matrices)
The transpose of A = a ij , denoted as A T , is defined by A = a ji .
Note that: the rows of A T are columns of A and the columns of A T are rows of A.
Definit
5.2 Permutations and Cofactors
Let A be an n n matrix and A a ij .
Three ways to compute the determinant of A:
(1) Pivot numbers:
(a) Reduce A to U u ij an upper triangular matrix.
(b) detA 1 k detU 1 k the product of u 11 u 22 u nn , where k is the numbe
3.5 Independence, Basis and Dimension
Definition: Vectors v 1 , . . . , v n are linearly independent if the only linear combination
c 1 v 1 . . . c n v n that gives 0 is having all zero coefficients: c 1 0, c 2 0, . . . , c n 0.
c1
AC 0.
Note that c 1 v
Math 530 - Fall, 2012 Review for Hour Exam 1
1. Vectors:
v1
v2
v1
v1
,
v2
,
v3
v2
vn
(ii) + w (iii) w
a. vector operations: (i) cv
v
v
computation and geometric interpretation of a vector operation
b. linear combination of vectors: av, av + bw, a
5.1 Determinant and Properties
Definition: Determinant:
Let A be an n n matrix and A = a ij . Let M ij be the n 1 n 1 matrix obtained from A by deleting
the ith row and the jth column and C ij = 1 i+j M ij . The determinant detA of A is a scalar and is de
5.3 Cramers Rule, Inverse Matrices and Volumns
Let A be an n n matrix and A a ij .
Cramers Rule to solve Ax b when detA 0:
Let B i be the n n matrix obtained from A by replacing the ith column of A by the vector b.
x1
detB i
. Then x i
, i 1, . . . , n.
3.3 The Rank and the Row Reduced Form
Let A be an m n matrix.
Definition: The rank of A, denoted as r, is the number of pivots.
the rank of A gives the true size of A.
Compute the rank of A using Gaussian-Elimination:
Gaussian-Elimination (downward): A U
3.2 The Nullspace of A and Solutions of Ax 0
Let A be an m n matrix.
Definition: The nullspace of A consists of all solutions to Ax 0.
NA x in R n ; Ax 0
NA is a subspace of R n .
Vector x in NA if and only if x is a solution of Ax 0
Solve Ax b using
(
3.4 The Complete Solution to Axb
Let A be an m n matrix, b be an m 1 vector.
Suppose Ax b is consistent (Ax b has a solution).
Let x be a solution of Ax b. Then x x p x N where
x p , called a particular solution, is a solution of Ax b; and
x N , a solutio
Math 530 - Fall, 2012 Hour Exam 2 Name: _Solutions_
1. (10pt) Let A and B be n n symmetric matrices. Prove or disprove each of the following matrices is also
symmetric. Show your work in detail to support your answer.
Know that A T A, and B T B.
(1) A 2 B