Matrices

$A$ is $m\times n$ matrix

Arithmetic

Matrix addition & multiplication

Definitions:

• Matrix addition - $A+B=C$ where $a_{ij}+b_{ij}=c_{ij}$
• Matrix multiplication - $A_{m\times n}{B}_{n\times p}=C_{m\times p}=[c_{ij}]$, where $c_{ij}$ is the dot product of the $i$th row of $A$ and the $j$th column of $B$
  • (3.4.3) $[A{]}_i^{r}B=[C{]}_i^r$ and $A[B{]}_j^c=[C{]}_j^c$
  • (3.4.4) $[A{]}_i^r=0⟹[C{]}_i^r=0$ and $[B{]}_j^c=0⟹[C{]}_j^c=0$

Properties:

• Commutative (addition) $A+B=B+A$
• Commutative (multiplication) see Commuting
• Associative (addition) $A+(B+C)=(A+B)+C$
• Associative (multiplication) $A(BC)=(AB)C$
• distributive (left) $A(B+C)=AB+AC$
• distributive (right) $(B+C)A=BA+CA$

Matrix–Vector Product

• $(A+B)\mathbf{v}=A\mathbf{v}+B\mathbf{v}$

• $A(\mathbf{u}+\mathbf{v})=A\mathbf{u}+A\mathbf{v}$

• $A(c\mathbf{u})=c(A\mathbf{u})$

• $A\mathbf{u}=A\mathbf{v}⟺A(\mathbf{u}-\mathbf{v})=0$

Row Echelon form (REF)

• (1.11.1) $\text{REF}(A)$ is row equivalence to $\text{RREF}(A)$
• (8.5.1) The non-zero rows of $\text{REF}(A)$ are a basis for $\text{row-space(A)}$.

Reduced Row Echelon form (RREF)

• Uniqueness: Each matrix is row equivalent to one and only one reduced echelon matrix.

Row equivalence

Row equivalence is an equivalence relation

Equivalent Definitions:

• $A$ and $B$ are row equivalent
• It is possible to transform $A$ into $B$ by a sequence of elementary row operations
• (q7.5.12) $\text{row-space}(A)=\text{row-space}(B)$
• $\text{null}(A)=\text{null}(B)$
• There exists an invertible matrix $P$ such that $A=PB$.

Properties: If $A$ and $B$ are row equivalent matrices, then:

• A given set of column vectors of $A$ is linearly independent if and only if the corresponding column vectors of $B$ are linearly independent.
• A given set of column vectors of $A$ forms a basis for the column space of $A$ if and only if the corresponding column vectors of $B$ form a basis for the column space of $B$.
• $A$ and $B$ have the same rank
• (4.2.2) $\det A=0⟺\det B=0$

Fundamental Spaces

Row space

Definitions:

• $\text{row-space}(A)=$

Theorems:

• $\text{row-space}(A+B)\subseteq \text{row-space}(A)+\text{row-space}(B)$

Column space

Theorems:

• (9.8.7a) $\text{column-spcae}(BA)\subseteq \text{column-spcae}(B)$
• (9.8.7c) $\text{column-spcae}(BA)=B(\text{column-spcae}(A))$

Null space

Definitions:

• $\text{null}(A)=\{v\mid Av=0\}$. (in the book it’s notated as $P(A)$ (!!) )

Theorems:

• (9.8.7b) $\text{null}(A)\subseteq \text{null}(BA)$

Rank

Notation

$\text{rank}(A)=\rho (A)$

Definitions:

• (d8.5.4) $\rho (A)=\dim (\text{row-space(A)})=\dim (\text{column-space(A)})$
• $\rho (A)$ is the number of linearly independent rows
• $\rho (A)$ is the number of linearly independent columns
• $\rho (A)=\dim (\text{Im}(T_A))$
• $\rho (A)=n-\dim (\text{null}(A))$
• $\rho (A)$ is the number of the non-zero rows of $\text{REF}(A)$
• $\rho (A)$ is the number of pivots in $\text{RREF}(A)$

Theorems:

• (q8.5.4) $\rho (A)=\rho (A^t)$

• (q8.5.5) $\rho (A_{m\times n})\le \min \{m,n\}$

  • if $\rho (A)=\min \{m,n\}$ then $A$ has full rank.
    • if $\rho (A)=n$ then $A$ has full column rank. ($f(\mathbf{x})=A\mathbf{x}$ is injective)
    • if $\rho (A)=m$ then $A$ has full row rank. ($f(\mathbf{x})=A\mathbf{x}$ is surjective) (A’s rows are linearly indepndenttodo )
  • Otherwise if $\rho (A)<\min \{m,n\}$, then $A$ has rank deficient.

• (q8.5.6) $\rho (AB)\le \min \{\rho (A),\rho (B)\}$

• if $A_n$ is invertible matrix, then

  • (q8.5.7a) $\rho (BA)=\rho (B)$ for any matrix $B_{m\times n}$
  • (q8.5.7b) $\rho (AB)=\rho (B)$ for any matrix $B_{n\times m}$

• (8.6.1) Rank–nullity theorem $\rho (A)+\dim (\text{null}(A))=n$

• $A=0⟺\rho (A)=0$

• Row equivalent matrices have the same ranktodo

• todo $\rho (A+B)\le \rho (A)+\rho (B)$

• (8.3.4a+8.6.1) $\text{null}(AB)\subseteq \text{null}(B)⟹\rho (B)\le \rho (AB)$

• rank of square matrix: let $A,B$ are square matrices of order $n$, then:

  • (q8.5.8a) $\rho (A)=n⟺\det (A)\ne 0$
  • (q10.5.3) Sylvester’s inequality $\rho (A)+\rho (B)\le \rho (AB)+n$
  • $\rho (B)-\dim (\text{null}(A))\le \rho (AB)$. (from Sylvester’s inequality and Rank–nullity theorem)
  • $AB=0⟹\rho (A)+\rho (B)\le n$

Nullity

• (8.6.1) $\mathrm{nullity}(A)=\dim (\mathrm{null}(\mathrm{A}))=n-\rho (A)$

Transformation matrix

Matrix Representations of Linear Transformation

• (d10.1.1) $T:V\to W$, and $B=(v_1,\dots ,v_n)$ and $C=(w_1,\dots ,w_n)$ are bases of $V$ and $W$. (respectively) $[T{]}_C^B={\left[\begin{array}{ccc}|& & |\\ [T(v_1){]}_C& \cdots & [T(v_n){]}_C\\ |& & |\end{array}\right]}_{m\times n}$

Transpose

• Notation: $A^t$, $A^{\top }$
• (3.2.4) $(A^t{)}^t=A$
• $(A+B{)}^t=A^t+B^t$
• (3.4.5) $(AB{)}^t=B^t{A}^t$
• $(cA{)}^t=c{A}^t$

Equivalence

This term is beyond the scope of the course.

Matrix equivalence is an equivalence relation on the space of rectangular matrices.

Two $m\times n$ matrices $A$ and $A^{\prime }$ are called equivalent if $A^{\prime }=P^{-1}AQ$for some $n\times n$ matrix $Q$ and $m\times m$ matrix $P$.

Equivalent matrices represent the same linear transformation $T:V\to W$ under two different choices of a pair of bases of $V$ and $W$, with $Q$ and $P$ being the change of basis matrices in $V$ and $W$ respectively.

• $B,C$ are old bases of $V,W$ (respectively)
• $B^{\prime },C^{\prime }$ are new bases of $V,W$ (respectively)
• $Q$ is change-of-basis $n\times n$ invertible matrix of $V$ from $B$ to $B^{\prime }$
• $P$ is change-of-basis $m\times m$ invertible matrix of $W$ from $C$ to $C^{\prime }$
• $A=[T{]}_C^B$ is the transformation matrix by the old bases
• $A^{\prime }=[T{]}_{C^{\prime }}^{B^{\prime }}$ is the transformation matrix by the new bases

Theorems

• The following statements are equivalent:

  • For each $\mathbf{b}$ in ${\mathbb{R}}^m$, the equation $A\mathbf{x}=\mathbf{b}$ has a solution
  • Each $\mathbf{b}$ in ${\mathbb{R}}^m$ is a linear combination of the columns of $A$
  • The columns of $A$ span ${\mathbb{R}}^m$
  • $A$ has a pivot postion in every row

• The following statements are equivalent:

  • The equation $A\mathbf{x}=\mathbf{b}$ has a unique least-squares for each $\mathbf{b}$ in ${\mathbb{R}}^m$
  • The columns of $A$ are linearly indepndent
  • The matrix $A^{t}A$ is invertible

• $AB=0⟺\text{column-space}(B)\subseteq \text{null}(A)$

Square Matrices

• Let $A$ be an $n\times n$ square matrix.
• $A=[T{]}_B$ where $B$ is a basis of $V$

theorems:

• $A^2=0⟹\text{column-space}(A)\subseteq \text{null}(A)$
• $\text{row-space}(A)=\text{column-space}(A)⟹\text{column-space}(A)\oplus \text{null}(A)={\mathbb{R}}^n$ (by 9.3.7, 12.3.1, 12.3.2a, e2023a85q1a)

Invertibility

• Theorem 3.10.6: Let $A$ be a $n$-ordered square matrix over a field $F$. The following statements are equivalent:
  • $A$ is an invertible matrix
  • $A$ can be expressed as a finite product of elementary matrices.
  • There exists a $B$ such that $BA=I$
  • There exists a $B$ such that $AB=I$
  • There exists a $B$ such that $AB=BA=I$, (in such case $A^{-1}=B$, and $B^{-1}=A$) ()
  • $A$ is row-equivalent to $I$.
  • $A$ is column-equivalent to $I$.
  • $\text{RREF}(A)=I$
  • The columns of A are linearly independent.
  • The rows of A are linearly independent.
  • The columns of A span $F^n$
  • The rows of A span $F^n$
  • The columns of A is a basis $F^n$
  • The rows of A a basis $F^n$
  • $A^t$ is an invertible matrix (in such case $(A^t{)}^{-1}=(A^{-1}{)}^t$) (3.8.4b)
  • (4.4.1, q10.7.7 for l.t.) The determinant of A is non-zero: $\det A\ne 0$
  • (4.4.1, and q11.3.1) The number $0$ is not an eigenvalue of $A$.
  • (q8.5.8b) $A$ has a full rank: $\rho (A)=n$
  • (10.5.1, and 9.6.2) $\text{null}(A)=\{0\}$
  • $(\text{null}(A){)}^{\perp }=F^n$
  • $\dim (\text{null}(A))=0$
  • The linear transformation mapping $\mathbf{x}$ to $A\mathbf{x}$ is surjective; that is, the equation $A\mathbf{x}=\mathbf{b}$ has at least one solution for each $\mathbf{b}$ in $F^n$.
  • The linear transformation mapping $\mathbf{x}$ to $A\mathbf{x}$ is injective; that is, the equation $A\mathbf{x}=\mathbf{b}$ has at most one solution for each $\mathbf{b}$ in $F^n$.
  • The linear transformation mapping $\mathbf{x}$ to $A\mathbf{x}$ is bijective; that is, the equation $A\mathbf{x}=\mathbf{b}$ has exactly one solution for each $\mathbf{b}$ in $F^n$. ($A\mathbf{x}=\mathbf{b}⟹\mathbf{x}=A^{-1}\mathbf{b}$
  • $A\mathbf{x}=0⟹\mathbf{x}=0$

Properties:

• for $A_n$ invertible matrix
  • (3.8.3)
    • (left-cancellable) $AB=AC⟹B=C$
    • (right-cancellable) $BA=CA⟹B=C$
  • (3.8.4d) if $s\ne 0$, then $sA$ is also invertible. (in such case $(sA{)}^{-1}=\frac{1}{s}{A}^{-1}$)
  • (q8.5.7a) $\rho (BA)=\rho (B)$ for any matrix $B_{m\times n}$
  • (q8.5.7b) $\rho (AB)=\rho (B)$ for any matrix $B_{n\times m}$
• if $A$ and $B$ are invertible, (in order $n$)
  • $A$ and $B$ are row equivalent
  • (3.8.4c) $AB$ is also invertible. and $(AB{)}^{-1}=B^{-1}{A}^{-1}$

Theorems:

• (4.5.2) $A,B$ are square matries, and $AB=I$, then:
  • $A$ and $B$ are both invetible
  • $A^{-1}=B$
  • $B^{-1}=A$
  • $AB=BA=I$
• (q3.10.2) $A,B$ are invertible, if and only if, $AB$ is invertible
• $A,B,C$ are square matries
  • if $AB-AC$ is invertible and $AB$ is singular, then $B$ is singular

Procedure: determine whether a square matrix A is invertible and, if so, find A^{−1}:

• Form the augmented matrix $[A|I_n]$ and put it into RREF.
• If the RREF has the form $[I_n|B]$, then $A$ is invertible and $B=A^{-1}$.
• else, if the matrix in the left half of the RREF is not $I_n$, then $A$ is singular.

WolframAlpha inverse [matrix]

Elementary matrix

• $A$ is called an elementary matrix if it can be obtained from an identity matrix by performing a single elementary row operation.
• Every elementary matrix is invertible, and the inverse is also an elementary matrix.

Determinant

• Notation: $\det A=|A|$

• Cofactor Expansion: $\det (A)={\sum }_{j=1}^n(-1{)}^{i+j}{a}_{ij}\det (A_{ij})$

  • $a_{ij}$ is the entry of the $i$th row and jth column of $A$
  • $A_{ij}$ is the submatrix obtained by removing the $i$th row and the $j$th column of $A$
  • Here $j$ is variable and $i$ is some consonent, but the opposite is also possible.
  • $M_{ij}=\det (A_{ij})$ is $i,j$ minor of $A$.
  • $C_{ij}=(-1{)}^{i+j}{M}_{ij}$ is cofactor of entry $a_{ij}$

• Theorems:

  • (4.3.1) $\det (A)=\det (A^t)$
  • (4.2.2) if $A$ has zero row/column, then $\det A=0$
  • (4.3.5) if $A$ has two equal rows (or colmuns), then $\det A=0$
  • (4.5.1) Multiplicative Property: $\det (AB)=\det (A)\det (B)$
  • (by 4.5.1) $\det (AB)=\det (BA)$
  • (4.5.3) $\det (A^k)=\det (A{)}^k$
  • (4.5.4) $\det (A^{-1})=\frac{1}{\det (A)}$
  • (4.4.1) $\det A\ne 0⟺A\text{ is invertible}$
  • (q4.4.4) if the sum of each of $A$s rows is zero, then $\det A=0$
  • (4.3.8) if $A_n=[a_{ij}]$ is a triangular matrix, then $\det A=a_{11}{a}_{22}\cdots a_{nn}$
  • (10.7.3) similar matrices have the same determinant
  • (q11.3.1) $\det A=0$, if and only if, $\lambda =0$ is eigenvalue of $A$
  • Row Operations
    • (4.3.6) If a multiple of one row of $A$ is added to another row to produce a matrix $B$, then $\det B=\det A$
    • (4.3.2) If two rows of $A$ are interchanged to produce $B$, then $\det B=-\det A$
    • (4.3.3) if one row of $A$ is multiplied by $k$ to produce $B$, then $\det B=k\cdot \det A$
  • (q4.3.3b) Homogeneity: $\det (tA)=t^n\det A$
  • $\det A$ is equal to the product of its eigenvalues (see q11.4.7)
  • (q4.3.10) determinant of an odd dimension anti-symmetric matrix is zero
  • (4.3.4) Let A, B, and C be n x n matrices that differ only in a single row, say the rth, and assume that the rth row of C can be obtained by adding corresponding entries in the rth rows of A and B. Then $\det (C)=\det (A)+\det (B)$ The same result holds for columns.

Procedure: computing the detrminant

1. convert $A$ into an upper triangular matrix $B$ via row operations of switching rows, and adding multiplication of another row. (but NOT scaling of row by scalar)

• Note: if during the row operations we find zero-row, then $\det A=0$

2. let $k$ be the number of times two rows are switched
3. $\det A=(-1{)}^k{b}_{11}\cdot \cdot \cdot b_{nn}$

Trace

• $\text{tr}(A)={\sum }_{i=1}^n{a}_{ii}=a_{11}+a_{22}+\cdots +a_{nn}$
• $\text{tr}A$ is the sum of its eigenvaluestodo
• (10.7.6) $\text{tr}(AB)=\text{tr}(BA)$
• (10.7.5) similar matrices have the same trace
• $\text{tr}(A+B)=\text{tr}(A)+\text{tr}(B)$
• $\text{tr}(cA)=c\text{tr}(A)$todo
• $\text{tr}(A)=\text{tr}(A^t)$todo

Characteristic polynomial

$p(t{)}_A=\det \left(tI-A\right)$ Properties:

• The characteristic polynomial is a monic polynomial of degree $n$
• (q11.5.4) The coefficient of $t^n$ is $1$
• (q11.4.6) The coefficient of $t^{n-1}$ equals $\text{tr}(-A)=-\text{tr}(A)$
• (q11.4.7) The free coefficient equals $\det (-A)=(-1{)}^n\det (A)$
• The characteristic polynomial of $A_2$ is $t^2-\text{tr}(A)t+\det (A)$

Eigenvalues

Equivalent definitions of eigenvalue.

• $\lambda$ is an eigenvalue of $A$
• (d11.3.1) There exists a non-zero vector $v$ such that $Av=\lambda v$.
  • (in such case, $v$ is called an eigenvector of $A$ that related to the eigenvalue $\lambda$)
• $(A-\lambda I)$ is singular
• $\text{rank}(A-\lambda I)<n$
• $(\lambda I-A)v=0$ has nontrivial solutions, i.e. $\text{null}(\lambda I-A)\ne \{0\}$
• (11.4.1) The characteristic equation $\det (\lambda I-A)=0$
• $\lambda$ is a root of the characteristic equation $\det (xI-A)=0$
• $\lambda$ is an eigenvalue of $T_A$

Theorems:

• Similar matrices have the same eigenvalues (11.3.3), the same characteristic polynomial (11.4.3), and the same algebraic multiplicities of eigenvalues (todo )
• The sum of eigenvalues of $A$ equals to $\text{tr}A$todo
• The product of eigenvalues of $A$ equals to $\text{det}A$todo
• (q11.3.2a) if $\lambda$ is an eigenvalue of $A$, then for each $\mu$, $\mu \lambda$ is an eigenvalue of $\mu A$
• (q11.3.2b) if $\lambda$ is an eigenvalue of $A$, then , ${\lambda }^k$ is a eigenvalue of $A^k$. (for each natural $k$ )
• The eigenvalues of a triangular matrix equal the values on its diagonal.
• (q11.3.5b) The eigenvalues of diagonal matrix $\text{diag}({\lambda }_1,\dots ,{\lambda }_n)$, are ${\lambda }_1,\dots ,{\lambda }_n$
• (q11.3.5b, q11.3.6) The eigenvalues of diagonalizable matrix (that similar to $\text{diag}({\lambda }_1,\dots ,{\lambda }_n)$) are ${\lambda }_1,\dots ,{\lambda }_n$
• if $A^n=I$ for some natural $n$, then $A$ has at most the eigenvalues $1,-1$ (todo by q11.2.4)
• (11.2.6) $A$ has at most $n$ distinct eigenvalues
• (4.4.1+q11.3.1+left-multiple with A) if $A$ is invertible, then $\lambda$ is an eigenvalue of $A$, if and only if, ${\lambda }^{-1}$ is an eigenvalue of $A^{-1}$. (with the same eigenvectors)

Eigenvectors

Definitions of eigenvector. The following statements are equivalent:

• $v$ is an eigenvector of $A$ that related to $\lambda$
• (d11.3.1) $v$ is non-zero vector in ${\mathbb{R}}^n$ such that $Av=\lambda v$
• (11.3.2) $[v{]}_B$ is an eigenvector of $T_A$ that related to $\lambda$

Eigenbasis

• An eigenbasis of $A$, is a basis of $F^n$ consisting of eigenvectors of $A$.
• if ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ are distinct eigenvalues of a matrix $A$ with corresponding eigenspaces, ${\mathcal{E}}_1,{\mathcal{E}}_2,\dots {\mathcal{E}}_k,$ spanned by bases ${\mathcal{B}}_1,{\mathcal{B}}_2,\dots {\mathcal{B}}_k,$ respectively, then the union ${\mathcal{B}}_1\cup {\mathcal{B}}_2\cup \cdots \cup {\mathcal{B}}_k$ is linearly independent set of eigenvectors of $A$. thereby if the size of the union is $n$, then is also an eigenbasis of $A$

Eigenspace

Definitions of the eigenspace of $A$ associated with its eigenvalue $\lambda$.

• $\{\mathbf{v}\mid (A-\lambda I)\mathbf{v}=0\}$
• $\{\mathbf{v}\mid A\mathbf{v}=\lambda \mathbf{v}\}$
• $\{\text{eigenvectors corresponding to }\lambda \}\cup \{0\}$
• $\text{null}(\lambda I-A)$

Algebraic & geometric multiplicity

• $\lambda$ is an eigenvalue of $A$

  • (d11.5.2) The algebraic multiplicity of $\lambda$ is:
    • the multiplicity of $\lambda$ as a root of the characteristic equation
    • the highest power of $(x-\lambda )$ that divides the characteristic polynomial of $A$
  • (q11.5.2) The geometric multiplicity of $\lambda$, is:
    • the dimension of the eigenspace corresponding to $\lambda$,
    • $\dim (\text{null}(\lambda I-A))=n-\text{rank}(\lambda I-A)$
  • todo if $A$ is diagonalizable, then the geometric and algebraic multiplicity of $\lambda$ is the number that $\lambda$ appears in the diagonalization of $A$
  • (11.5.3, q11.5.3) $1\le$ the geometric multiplicity $\le$ the algebraic multiplicity

• finding the algebraic multiplicity of eigenvaluetodo

• finding the geometric multiplicity of eigenvaluetodo

Procedure: Finding Eigenvalues and Eigenvectors

1. First, find the eigenvalues $\lambda$ of $A$ by solving the characteristic equation $\det \left(\lambda I-A\right)=0$.
2. For each $\lambda$, find the basic eigenvectors $v\ne 0$ by finding the basic solutions to $\left(\lambda I-A\right)v=0$.

To verify your work, make sure that $Av=\lambda v$ for each $\lambda$ and associated eigenvector $v$.

Similarity

Similarity is an equivalence relation on the space of square matrices.

$A$ and $B$ are square matrices

Definitions of similarity. The following statements are equivalent:

• $A$ and $B$ are similar
• (d10.7.1) There exists an invertible matrix $P$ such that $A=P^{-1}BP$
  • $P$ being the change of basis matrix
• (10.7.2) $A$ and $B$ represent the same linear transformation. (possibly different bases)

Theorems:

• (q10.7.8) zero matrix is similar only to itself. identify matrix is similar only to itself.
• todo to show that two matrices are similar, show that are similar to the same diagonal matrix
• todo let $A$ and $B$ are diagonalizable, and they both have the same eigenvalues, then they’re similar (because similarity is transitive)
• todo let $A$ and $B$ are diagonalizable, and they both have the same characteristic polynomial, then they’re similar (because similarity is transitive)

Properties:

• If the matrices $A$ and $B$ are similar, then
  • $A=I⟺B=I$todo
  • $A=0⟺B=0$todo
  • $A$ is invertible, if and only if $B$ is also invertibletodo
  • $\det (A)=\det (B)$ (10.7.3)
  • $\text{tr}(A)=\text{tr}(B)$ (10.7.5)
  • $A$ and $B$ have the same eigenvalues (11.3.3)
  • $A$ and $B$ have the same algebraic multiplicities of eigenvaluestodo
  • $A$ and $B$ have the same characteristic polynomial (11.4.3)
  • $\dim (\text{null}(A))=\dim (\text{null}(B))$
  • $\text{rank}(A)=\text{rank}(B)$

Triangular matrix

Properties:

• if $A_n=[a_{ij}]$ is a triangular matrix, then
  • $\det A=a_{11}{a}_{22}\cdots a_{nn}$ (4.3.8)
  • the eigenvalues of $A$ are $a_{11},a_{22},\dots ,a_{nn}$
    • each eigenvalue occurs exactly k times on the diagonal, where k is its algebraic multiplicity
  • the characteristic polynomial of $A$ is $p_A=(x-a_{11})(x-a_{22})\cdots (x-a_{nn})$

Diagonal matrix

Diagonal equivalent definitions.

• $A$ is a diagonal matrix
• $A$ is both upper- and lower-triangular
• see q11.1.1

Properties:

• Addition: $\text{diag}(a_1,\dots ,a_n)+\text{diag}(b_1,\dots ,b_n)=\text{diag}(a_1+b_1,\dots ,a_n+b_n)$
• Multiplication $\text{diag}(a_1,\dots ,a_n)\text{diag}(b_1,\dots ,b_n)=\text{diag}(a_1{b}_1,\dots ,a_n{b}_n)$
• Powers of a matrix $(\text{diag}(a_1,\dots ,a_n){)}^k=\text{diag}(a_1^k,\dots ,a_n^k)$
• $\text{diag}(a_1,\dots ,a_n)\text{ is invertible}⟺a_1,\dots ,a_n\ne 0$. in such case $\text{diag}(a_1,\dots ,a_n{)}^{-1}=\text{diag}(a_1^{-1},\dots ,a_n^{-1})$
• A diagonal matrix is symmetric.
• $\det (\text{diag}(a_1,\dots ,a_n))=a_1\cdots a_n$
• the rank of a diagonal matrix is simply the number of nonzero entries (the eigenvalues)

Diagonalizable

• Diagonalizable definition. The following statements are equivalent.

  • $A$ is a diagonalizable matrix
  • (d11.3.4) There exists an $n\times n$ invertible matrix $P$, such that $P^{-1}AP$ is a diagonal matrix
  • (d11.3.4) $A$ is similar to a diagonal matrix
  • (11.3.7) $A$ has $n$ linearly independent eigenvectors. (they are $P$‘s columns. that are $v_1,\dots ,v_n$, and $v_i$ is a eigenvector of $A$ that’s related to the eigenvalue ${\lambda }_i$. and $P^{-1}AP=\text{diag}({\lambda }_1,\dots ,{\lambda }_n)$ )
  • (q11.3.7) $F^n$ has a basis that consists of eigenvectors of $A$
  • (11.3.5) $T_A:V\to V$ is diagonalizable
  • (q11.4.10) $A^t$ is diagonalizable
  • (11.5.4’)
    • (i) the characteristic polynomial factors completely into linear factors. and
    • (ii) the geometric multiplicity of every eigenvalue is equal to the algebraic multiplicity
  • todo The sum of the dimensions of the eigenspaces equals to $n$

• (11.3.6) if $A$ has $n$ distinct eigenvalues, then $A$ diagonalizable

• todo $A^k=P{D}^k{P}^{-1}$, ($D$ is a diagonal matrix)

Symmetric matrix

• (d3.2.6) $A\text{ is symmetric}⟺A=A^t$
• (q3.2.3) $A\text{ is symmetric}⟹A\text{ is square}$
• (q3.2.4) $A\text{ is diagonal}⟹A\text{ is symmetric}$
• (q3.2.4) sum of symmetric matries is symmetric matrix
• (q3.4.6) $A$ and $B$ are symmetric matries, then $AB\text{ is symmetric}⟺AB=BA$
• (q4.3.10) $A\text{ is anti-symmetric}⟺A^t=-A$
  • if $A_n$ is anti-symmetric, and $n$ is odd, then $\det A=0$

Change of Basis matrix (Transition matrix)

also change-of-coordinates matrix

• (d8.4.6) Let $B=(v_1,\dots ,v_n)$ and $C=(u_1,\dots ,u_n)$ bases of $V$. if \begin{align} u_{1} &= a_{11}v_{1}+\dots+a_{n1}v_{n} \\ \vdots \notag \\ u_{n} &= a_{1n}v_{1}+\dots+a_{nn}v_{n} \\ \end{align}then $P=\left[\begin{array}{ccc}{a}_{11}& \dots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{n1}& \dots & a_{nn}\end{array}\right]=\left[\begin{array}{ccc}|& & |\\ [u_1{]}_B& \cdots & [u_n{]}_B\\ |& & |\end{array}\right]$ is the transition matrix from basis $B$ to basis $C$.
• the transition matrix from basis $B$ to basis $C$ is the matrix that its columns are the coordinate vectors of the $C$ vectors by $B$.
• $\forall v\in V:P[v{]}_C=[v{]}_B,[v{]}_C=P^{-1}[v{]}_B$
• (8.4.9) $P^{-1}$ is the transition matrix from $C$ to $B$
• Transition matrix is square matrix of $n$-order, where $n=\dim V$
• (by 8.4.5) Transition matrix is invertible matrix

Theorems:

• (8.4.8) if $A$ is square matrix, and $[v{]}_B=P[v{]}_C$ for each $v\in V$, then $P$ is the transition matrix from $B$ to $C$.
• (10.6.1) $T:V\to V$ and $B$ and $C$ are bases of $V$. if $P$ transition matrix from $B$ to $C$, then $[T{]}_C=P^{-1}[T{]}_{B}P$. (or symmetrically $[T{]}_B=P[T{]}_C{P}^{-1}$)
  • transition matrix from $B$ to $C$ is $[I{]}_B^C$, therefore $[T{]}_C=[I{]}_C^B[T{]}_B[I{]}_B^C$

Finding the transition matrix from an old basis to a new basis

1. Form the partitioned matrix $[\text{new basis}|\text{old basis}]$ in which the basis vectors (or coordinate vectors) are in column form.
2. Use elementary row operations to reduce the matrix in Step 1 to RREF.
3. The resulting matrix will be $[I|\text{transition matrix from old to new}]$ where $I$ is an identity matrix.
4. Extract the matrix on the right side of the matrix obtained in Step 3.

Transition matrix from a basis B to the standard basis

• if $B=(v_1,v_2,\dots ,v_n)$, then $[v_1|v_2|\dots |v_n]$ is the transition matrix from $B$ to the standard basis

Orthogonality

todo Orthogonal matrix - This is not taught in the course.

Commuting

• (d3.6.2) $A$ and $B$ are said to commute if $AB=BA$
• (3.6.3) $(tI)A=A(tI)$
• $A$ and $B$ share the same $n$ independent eigenvectors if and only if $AB=BA$.