LA筆記:(六)Diagonal,Triangular and Symmetric matrices

前言

本篇關於對角線矩陣、三角形矩陣、對稱矩陣的定義及他們的性質。這三種矩陣都是建立在方陣的前提下，是方陣的一種特殊形式。

Diagonal matrix(對角線矩陣)

Def:

A square matrix D is called a diagonal matrix if Dij = 0 for all i≠j.

除了對角線上，其他元素都為0的方陣叫做對角線矩陣。

Properties of Diagonal matrix:

• 對角線矩陣做n次冪運算的結果，為對角線上元素分別做n次冪運算，其他位置元素依舊為0.

• 對角線矩陣的可逆判斷條件：A n-by-n diagonal matrix D is invertible if and only if Dii≠0 for all 1<= i <=n. 對角線上元素全部不為0時，對角線矩陣可逆。對角線可逆則其對角線元素全部不為0.

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Triangular matrix(三角形矩陣)

Def:

A square matrix A is called a upper triangular matrix if Aij = 0 for all i > j.

A square matrix A is called a lower triangular matrix if Aij = 0 for all i < j.

三角形矩陣可逆判斷條件

A triangular matrix is invertible if and only if its diagonal entry are all nonzeros.與對角線矩陣可逆判斷的條件相同。

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Symmetric(對稱矩陣)

Def:

A square matrix A is said to be symmetric if A = A^T (Aij = Aji for all i,j)

轉置矩陣等於自身的方陣稱謂對稱矩陣。

對稱矩陣一些性質

if A and B are symmetric matrices of the same size and k is a scalar, then

<a>A^T is symmetric <b>A+B and A-B are symmetric <c>kA is symmetric

注意：A,B兩方陣分別對稱但矩陣（A·B）不一定對稱

(A·B)^T = B^T · A^T = BA ≠ AB

兩對稱矩陣的乘積矩陣仍對稱的條件

The product of two symmetric matrices is symmetric if and only if the matrices are commutative.

只有在兩方陣可以交換的條件下，其乘積矩陣才對稱。

可逆對稱矩陣的逆矩陣對稱。

If A is an invertible and symmetric matrix, then A^-1 is symmetric.

Proof:

(A^-1)T = (A^T)-1 = A^-1

證明思路：套用對稱矩陣的定義（轉置矩陣為本身的方陣）證明

用普通矩陣構造出對稱矩陣的方法

let A be a matrix(need not square), then A · A^T and A^T · A is symmetric.

proof:

(A·A^T)T = (A^T)T · A^T = A · A^T and (A^T · A)^T = A^T · (A^T)T = A^T · A

證明思路：套用對稱矩陣的定義（轉置矩陣為本身的方陣）證明