linear algebra module 3

determinant

• real number

• related to invertibility and volume

properties of determinant

1. doing a row/column replacement on A does not change determinant

2. scaling a row/column scales the determinant by c

3. swapping two rows/columns multiplies determinant by −1.

row operations change the value of the determinant

A square matrix is invertible if …

if and only if det(A) =/ 0; in this case, det(A⁻¹)=1det(A).

det(AB) = 

det(A) * det(B)

for any SQUARE matrix, det(A^T) =

det(A)

The parallelepiped defined by a set of vectors has zero volume

if the matrix with rows v1,v2,...,vn has zero determinant.

area = | det(v1 v2) * det(A) |

the zero matrix is diagonalizable

eigenvalue of 0, alg mult 2

unique eigenspace for

each eigenvalue

*spaces may have multiple vectors

all eigenspaces together = dim of matrix

steady state vector

• eigenvalue corresponding to pivot 1

when A is singular

det = 0, eigenvalue of 0

area of parallelogram spanned by vectors a,b equals

area spanned by vectors a, ca + b

volume of a parallelepiped spanned by square matrix A equals

|det A|

probability vector

• vector with nonnegative elements that equal 1

• cannot be a subspace

stochastic matrix

square matrix with probability vectors as columns

• considered regular if P^k entries are all greater than 0 for some k (matrix P may have 0 entries)

• markov chains converge to UNIQUE steady state w POSITIVE entries when matrix is regular

• A STEADY STATE EXISTS FOR ANY STOCHASTIC MATRIX (so eigenvalue 1)

markov chains

a sequence of probability vectors such that X k+1 = Pxk

to calculate steady state vector:

(P - I) q = 0

matrix transpose properties

same det, same polynomial, same eigenvalues

*diff eigenvectors

zero vector

NEVER an eigenvector

diagonal matrices

• all diagonal matrices are diagonalizable

• all powers of a diagonal matrix are diagonalizable

an nxn matrix with n distinct eigenvalues will span

R^n

imaginary

• must be in conjugate pairs

• corresponding imaginary eigenvectors

triangular matrices

diagonal values are eigenvalues

similar matrices

• same rank, det, polynomial, eigenvalues

• P, B, P^-1

• different eigenvectors

diagonalization: similar to a diagonal matrix B

** matrices with the same eigenvalues are not necessarily similar

if AB has eigenvalue 0

BA must also