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Flashcards in Algebra I Deck (14):

Equivalent matrices

š“,šµ āˆˆ Kįµā‹…āæ are equivalent if there exists invertible matrices P āˆˆ Kįµā‹…įµ, Q āˆˆ Kāæā‹…āæ s.t. šµ = Pš“Q.



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


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).


Similar matrices

š“,šµ āˆˆ Kāæā‹…āæ are similar if there exists invertible matrix P āˆˆ Kāæā‹…āæ s.t. šµ = Pā»Ā¹š“P.


Characteristic polynomial

The char polynomial of an nƗn matrix š“ is
p_š“(š‘„) = det(š“ - š‘„Iā‚™)
where Iā‚™ is the nƗn identity matrix.


Jordan chain

A Jordan chain of length k is a sequence of non-zero vectors vā‚,...,vā‚– āˆˆ Kāæā‹…Ā¹ that satisfies
Avā‚ = Ī»vā‚, Avįµ¢ = Ī»vįµ¢ + vįµ¢-ā‚, 2 ā‰¤ i ā‰¤ k, for some eigenvalue Ī» of š“ āˆˆ Kāæāæ.


Generalised eigenspace

Given T: V ā†’ V linear and Ī» āˆˆ K an eval of T, the generalised eigenspace of T corresponding to Ī» is
{š‘„ āˆˆ V | (T - Ī»šˆ)ā±(š‘„) = 0 for some i āˆˆ ā„¤}.


Jordan block

The Jordan block with eigenvalue Ī» of degree k is a kƗk matrix J_{Ī»,k} = (Ī³įµ¢ā±¼), s.t.
Ī³įµ¢,įµ¢ = Ī» for 1 ā‰¤ i ā‰¤ k,
Ī³įµ¢,įµ¢ā‚Šā‚ = 1 for 1 ā‰¤ i < k,
Ī³įµ¢ā±¼ = 0 for j ā‰  i, j ā‰  i+1.


Block sum of š“,šµ

Let š“ āˆˆ Kįµā‹…įµ, šµ āˆˆ Kāæā‹…āæ. š“ āŠ• šµ, the block sum of š“ and šµ, is the (m+n)Ɨ(m+n) matrix with block form
š“ 0ā‚˜,ā‚™
0ā‚™,ā‚˜ šµ


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. āˆƒ Jordan blocks Jā‚, ... Jā‚– s.t.
[ETE] = Jā‚ āŠ• ... āŠ• Jā‚–.


Cayley-Hamilton theorem

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


Jordan basis of š“ matrix

A Jordan basis for a matrix š“ āˆˆ Kāæā‹…āæ is a basis of Kāæā‹…Ā¹ which is a union of Jordan chains.


[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 āˆƒ! aįµ¢ā±¼, āˆ€ 1 ā‰¤ i ā‰¤ n, 1 ā‰¤ j ā‰¤ m, s.t. T(eā±¼) = āˆ‘_{i=1, m} aįµ¢ā±¼fįµ¢.
We write [FTE] = (aįµ¢ā±¼)įµ¢ā±¼ as the matrix of T wrt E and F.


Rules of powers in char and min polynomial

c_š“(š‘„) = āˆ_{š‘–=1, k} (šœ†įµ¢ - š‘„)āæā± where nįµ¢ is the number of times šœ†įµ¢ appears on the diag of š“'s JCF.
šœ‡_š“(š‘„) = āˆ_{š‘–=1, k} (x - šœ†įµ¢)įµā± where mįµ¢ is the length of the longest chain for šœ†įµ¢.