Lesson 0: Refresher on Matrices
Economics of Finance
School of Economic, UNSW
Copyright By PowCoder代写 加微信 powcoder
Review of matrix algebra
Matrices: A combination of vectors with the same dimension.
• square/non-square
• invertible/non-invertible
Operations:
• transpose
• addition/subtraction • multiplication
Matrix Operations
Some tips:
• Addition/subtraction needs to be between matrices with identical shape:
Am×n ± Bm×n
• Dimension of the line vector in front of × needs to match
with the dimension of the column vector after ×: Al×m × Bm×n
• Most times we need to transpose a matrix because of the above two rules.
Matrix inversion
Matrix inversion is closely related to linear independency.
Definition
A set of n-dimensional column vectors, {A1, A2, …Am} are linearly independent iff. the unique solution of the linear equation set:
λ1A1 +λ2A2 +…+λmAm =0
A square matrix, An×n, is invertible iff its column vectors, {A1, A2, …An} are linearly independent.
has the unique solution:
Why does it matter?
A square matrix is invertible iff an arbitrary linear system Ax = b
Existence and Uniqueness of Solution
Suppose there exists a λ ̸= 0 s.t.
λ1a1 +λ2a2 +…+λnan =0,
then we have two scenarios:
• No solution;
• Thereexistsxs.t. Ax=b,thenforanyα̸=0, xα =x+αλalsosatisfiesAxα =b.
The function either have no solution or infinite solutions. Number of linearly independent λ is called degree of freedom.
Interchangeable Concepts
For a square matrix An×n, the following statements are equivalent:
• invertible
• non-singular
• full rank
• linearly independent in columns
• determinant is non-zero
• an arbitrary function Ax = b has unique solution.
125 M=1 1 3
1 Square? Yes.
2 Linearly independent? No, third column is a linear combination of the first and second columns
→ C3 = C1 + 2C2.
3 Check determinant is zero
Thus, not invertible
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com