线性代数: Matrix Algebra

Let $A$ be a $m\times n$ matrix

$A(BC)=(AB)C$
$A(B+C)=AB+AC$
$(B+C)A=BA+CA$
$r(AB)=(rA)B=A(rB)$
$I_mA=A=AI_n$

The cancellation laws do not hold for matrix multiplication.

If $AB=BA$ , we say $A,B$ commute with one another. (In general, $AB\ne BA$ )

Transpose and Symmatric Matrix

The matrix $A$ is symmetric if $A=A^T$

$(A^T)^T=A$
$(A+B)^T=A^T+B^T$
For any scalar $r$ , $(rA)^T=rA^T$
$(AB)^T=B^TA^T$

Sub-Matrix

Suppose $A$ is a $m\times n$ matrix. Submatrix $A(i|j)$ is the $(m-1)\times (n-1)$ matrix obtained from $A$ by removing row $i$ and column $j$ .

Determinant of a Matrix

Theorem: Determinant for Matrix Operation

Suppose $A$ is a square matrix. Let $B$ be the matrix obtained from $A$ by swapping 2 rows or columns. Then

\det B=-\det A

Let $C$ be the square matrix obtain from $A$ by multiplying a single row or column by scalar $r$ , then

\det C=r\det A

If $A$ has 2 same rows or 2 same columns, then

\det A=0

Let $D$ be the matrix obtained from $A$ by multiplying a row/column by scalar $r$ and then adding it to another row/column, then

\det D=\det A

Singular Matrix

Singular Matrix

Suppose $A$ is square matrix. $A$ is singular iff $\det A=0$ .

Inversion

Suppose $A,B$ are $n\times n$ matrices, and $AB=BA=I_n$ , we say $A$ is an inverse of $B$

Non-singularity and Inversibility

Suppose $A$ is a square matric of size $n$ . The following are equivalent.

$A$ is non-singular
$A$ row-reduces to $I_n$
The columns of $A$ are linearly independent
$A$ is invertible

Suppose $A$ is non-singular. Then the unique solution to $\mathcal{LS}(A,\mathbf{b})$ is $A^{-1}\mathbf{b}$

Cramer’s Rule

Cross Products

$a\cdot (b\times c)=(a\times b)\cdot c$
$a\times (b\times c)=(a\cdot c)b - (a\cdot b)c$

Eigenvalue

An eigenvector of an $n\times n$ matrix $A$ is a non-zero vector $x$ such that $Ax=\lambda x$ for some scalar $\lambda$ , $\lambda$ is called eigenvalue.

$\lambda$ is an eigenvalue of $A$ iff the equation
$(A-\lambda I)x=0$
has a nontrivial solution.

Eigenvalue of Triangular Matrix

The eigenvalues of a a triangular matrix are on its main diagonal.

Diagonalizable Matrix

Square matrix $A$ is diagonalizable if

A=PDP^{-1}

for some invertible $P$ and some diagonal matrix $D$ .

Diagonalizable and Eigenvectors

A square matrix $A$ is diagonalizable iff $A$ has $n$ linearly independent eigenvectors. And also,

P=[v_1\, v_2\, \cdots\, v_n], D=\begin{bmatrix}\lambda_1&0&\dots&0\\0&\lambda_2&\dots&0\\\vdots&\vdots&&\vdots\\0&0&\dots&\lambda_n\end{bmatrix}