Chapter 1 Vectors

• The meaning of Column Vectors

Sometimes a vector in n-space R^n is written vertically rather than horizontally. Such a vector is called a column vector.

There are no different between vertically vectors and horizontally vectors.

• Theorem 1.1

• Dot (Inner) Product

Two vectors which are made a dot product (or inner product) must be the same dimensions, and what we get is just a number.

If u\dot v = 0, u and v are said to be orthogonal (or perpendicular).

• Theorem 1.2

• Norm(Length) of a Vetor

A vector u is called a unit vector if the norm of u is 1, or, equivalently, if u\dot u=1.

• Theorem 1.3 Cauchy-Schwarz inequality

• Throrem 1.4

Chapter 2 Algebra of Matrices

For a single element a_{ij}, i shows which row the element is in, j shows which column the element is in.

E.g. 'm \times n' means this matrix has m rows and n columns.

'A zero matrix always equals to another zero matrix' is true or false?

Is false, they are possibly don't have same size(same number of rows and same number of columns).

• Matrix Addition and Scalar Multiplication

• Theorem 2.1

• Matrix Multiplication

another example：

another example：

AB isn't equal to BA.

e.g.

• Theorem 2.2

• Transpose of a Matrix

相當(dāng)于將矩陣先順時(shí)針旋轉(zhuǎn)90度，再水平翻轉(zhuǎn)180度.

如果是sqaure matrix，那么就相當(dāng)于沿著對(duì)角線翻折?。。。。?/span>

• Theorem 2.3

check the conclusion (iv):

• Square Matrices

• Diagonal and Trace

-Diagonal

-Trace

If a matrix isn't square trace, it has no trace.

• Theorem 2.4

• Identity Matrix and Scalar Matrices

• Remark

• Powers of Matrices, Polynomials in Matrices

why we should use Identity matrix?

-- we can't plus a matrix and a number.

-- we must trans the number to a scalar matrix.

• Invertible (Nonsingular) Matrices

How to find a inverse matrix for a square matrix?

1.the matrix must be a square matrix

2.construction of a equation

3.find the solves of the equation

conclusion:

• Determinant

It's easy to find a inverse of A, then.

If the determinant is zero, then the matrix has no inverse.

Inverse of an n \times n Matrix

• Special Typres of Square Matrices

(special kinds of square matrices)

--Diagonal and Triangular Matrices

e.g.

三角矩陣分為上三角矩陣(upper triangular)和下三角矩陣(below triangular)：

• Theorem 2.5

Remark: A nonempty collection A of matrices is called an algebra (of matrices) if A is closed under the operations of matrix addition, scalar multiplication, and matrix multiplication. Clearly, the square matrices with a given order form an algebra of matrices, but so do the scalar, diagonal, triangular, and lower triangular matrices.

• Symmetric Matrices and skew-Symmetric Matrices

Definition of Symmetric Matrices ： A^T = A

Definition of skew-Symmetric Matrices ： A^T = - A

• Orthogonal Matrices

A transpose equals to A inverse.

• Normal Matrices

• Block Matrices

把矩陣劃分為幾個(gè)區(qū)域分別參與計(jì)算：

then we have：

• Square Block Matrices

Chapter3 System of Linear Equation

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