Flashcards in Algebra I Deck (14):

1

## Equivalent matrices

### š“,šµ ā Kįµā āæ are equivalent if there exists invertible matrices P ā Kįµā įµ, Q ā Kāæā āæ s.t. šµ = Pš“Q.

2

## Monic

### A polynomial with coeffs in a field K is monic if the coeff of the highest power is 1.

3

## Minimal polynomial of a matrix š“

### The min. polynomial of a matrix š“ (or corresponding linear operator T) is the unique monic polynomial Ī¼_š“(š„) of minimal degree s.t. Ī¼_š“(š“) = 0 (or Ī¼_š“(T) = 0).

4

## Similar matrices

### š“,šµ ā Kāæā āæ are similar if there exists invertible matrix P ā Kāæā āæ s.t. šµ = Pā»Ā¹š“P.

5

## Characteristic polynomial

###
The char polynomial of an nĆn matrix š“ is

p_š“(š„) = det(š“ - š„Iā)

where Iā is the nĆn identity matrix.

6

## Jordan chain

###
A Jordan chain of length k is a sequence of non-zero vectors vā,...,vā ā Kāæā
Ā¹ that satisfies

Avā = Ī»vā, Avįµ¢ = Ī»vįµ¢ + vįµ¢-ā, 2 ā¤ i ā¤ k, for some eigenvalue Ī» of š“ ā Kāæāæ.

7

## Generalised eigenspace

###
Given T: V ā V linear and Ī» ā K an eval of T, the generalised eigenspace of T corresponding to Ī» is

{š„ ā V | (T - Ī»š)ā±(š„) = 0 for some i ā ā¤}.

8

## Jordan block

###
The Jordan block with eigenvalue Ī» of degree k is a kĆk matrix J_{Ī»,k} = (Ī³įµ¢ā±¼), s.t.

Ī³įµ¢,įµ¢ = Ī» for 1 ā¤ i ā¤ k,

Ī³įµ¢,įµ¢āā = 1 for 1 ā¤ i < k,

Ī³įµ¢ā±¼ = 0 for j ā i, j ā i+1.

9

## Block sum of š“,šµ

###
Let š“ ā Kįµā
įµ, šµ ā Kāæā
āæ. š“ ā šµ, the block sum of š“ and šµ, is the (m+n)Ć(m+n) matrix with block form

š“ 0ā,ā

0ā,ā šµ

10

## Jordan basis of T map

###
Let T: V ā V be linear. A Jordan basis for T and V is a finite basis E of V s.t. ā Jordan blocks Jā, ... Jā s.t.

[ETE] = Jā ā ... ā Jā.

11

## Cayley-Hamilton theorem

### Let c_š“(š„) be the char polynomial of a matrix š“, then c_š“(š“) = 0.

12

## Jordan basis of š“ matrix

### A Jordan basis for a matrix š“ ā Kāæā āæ is a basis of Kāæā Ā¹ which is a union of Jordan chains.

13

## [FTE] matrix of T wrt two bases

###
Let T: V ā W be a linear map. Let E = (eā, ..., eā) be a basis of V, F = (fā, ..., fā) be a basis of W.

Then ā! aįµ¢ā±¼, ā 1 ā¤ i ā¤ n, 1 ā¤ j ā¤ m, s.t. T(eā±¼) = ā_{i=1, m} aįµ¢ā±¼fįµ¢.

We write [FTE] = (aįµ¢ā±¼)įµ¢ā±¼ as the matrix of T wrt E and F.

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