Interesting Linear Algebra Theorems

Diagonalizable & Minimal Polynomial

A matrix or linear map is diagonalizable over the field F if and only if its minimal polynomial is a product of distinct linear factors over F.

Characteristic Polynomial

Let A be an n\times n matrix. The characteristic polynomial of A, denoted by p_A(t), is the polynomial defined by \displaystyle p_A(t)=\det(tI-A).

Cayley-Hamilton Theorem

Every square matrix over a commutative ring satisfies its own characteristic equation:

If A is an n\times n matrix, p(A)=0 where p(\lambda)=\det(\lambda I_n-A).


Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, \det(\lambda I-A)
\begin{aligned}  \lambda\ \text{is an eigenvalue of }A&\iff\det(\lambda I-A)=0\\  &\iff \lambda\ \text{is a root of the characteristic polynomial}.  \end{aligned}

The solution space of (\lambda I-A)\mathbf{x}=0 is called the eigenspace of A associated with the eigenvalue \lambda. The eigenspace is denoted by E_\lambda.

Sum/Product of Eigenvalues
– The sum of all eigenvalues of A (including repeated eigenvalues) is the same as Tr(A) (trace of A, i.e. the sum of diagonal elements of A)
– The product of all eigenvalues of A (including repeated eigenvalues) is the same as \det(A).

Finding Least Squares Solution Review and Others

Rotation Matrix

The rotation matrix
\displaystyle  R=\begin{pmatrix}  \cos\theta & -\sin\theta\\  \sin\theta & \cos\theta  \end{pmatrix}
rotates points in the xy-plane counterclockwise through an angle \theta about the origin.

For example rotating the vector (1,0) 45 degrees counterclockwise gives us:
\displaystyle  \begin{pmatrix}  \cos 45^\circ & -\sin 45^\circ\\  \sin 45^\circ & \cos 45^\circ  \end{pmatrix}  \begin{pmatrix}  1\\  0  \end{pmatrix}  =  \begin{pmatrix}  \frac{\sqrt{2}}{2}\\  \frac{\sqrt{2}}{2}  \end{pmatrix}.

Finding Least Squares Solution

Given Ax=b (inconsistent system), solve
\displaystyle A^TAx=A^Tb instead to get a least squares solution of the original equation.


If we know a least squares solution \mathbf{u} of A\mathbf{x}=\mathbf{b}, we can find the projection \mathbf{p} of \mathbf{b} onto the column space of A by \displaystyle \mathbf{p}=A\mathbf{u}.

Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)

If A is a matrix with n columns, then \displaystyle rank(A)+nullity(A)=n.

(rank(A)=number of pivot columns,

nullity(A)=number of non-pivot columns.)

Linear Independence and the Wronskian
A set of vector functions \vec{f_1}(x), \dots, \vec{f_n}(x) from \mathbb{R} to \mathbb{R}^n is linearly independent in the interval (\alpha,\beta) if \displaystyle W[\vec{f_1}(x),\dots,\vec{f_n}(x)]\neq 0 for at least one value of x in the interval (\alpha,\beta).

Dot Product and Span Summary

Dot Product

\text{span}\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}=\{c_1\mathbf{u_1}+c_2\mathbf{u_2}+\dots+c_k\mathbf{u_k}\mid c_1,c_2,\dots,c_k\in\mathbb{R}\}=\text{set of all linear combinations of } \{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\}.

V\subseteq\mathbb{R}^n is a subspace of \mathbb{R}^n if
1) V=\text{span}\{\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}\} for some vectors \mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k}.
2) V satisfies the closure properties:

(i) for all \mathbf{u},\mathbf{v}\in V, we must have \mathbf{u}+\mathbf{v}\in V.

(ii) for all \mathbf{u}\in V and c\in\mathbb{R}, we must have c\mathbf{u}\in V.

3) V is the solution set of a homogeneous system.

(Sufficient to check either one of Condition 1, 2, 3.)

For V to be a subspace, zero vector \mathbf{0} must be in V. (Since for \mathbf{u}\in V, 0\in\mathbb{R}, we have 0\mathbf{u}\in V.)

Linear Independence and Dependence
\mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k} are linearly independent if the system \displaystyle c_1\mathbf{u_1}+c_2\mathbf{u_2}+\dots+c_k\mathbf{u_k}=0 has only the trivial solution, i.e. c_1=c_2=\dots=c_k=0.

If the system has non-trivial solutions, i.e. at least one c_i not zero, then \mathbf{u_1},\mathbf{u_2},\dots,\mathbf{u_k} are linearly dependent.

Gaussian Elimination Summary

Row echelon form (REF)
For each non-zero row, the leading entry is to the right of the leading entry of the row above.

E.g. \begin{pmatrix}  0 & \mathbf{1} & 7 & 2\\  0 & 0 & \mathbf{9} & 3\\  0 & 0 & 0 & 0  \end{pmatrix}

Note that the leading entry 9 of the second row is to the right of the leading entry 1 of the first row.

Reduced row echelon form (RREF)
A row echelon form is said to be reduced, if in each of its pivot columns, the leading entry is 1 and all other entries are 0.

E.g. \begin{pmatrix}  1 & 0 & 0 & 2\\  0 & 1 & 0 & 3\\  0 & 0 & 1 & 4  \end{pmatrix}

Elementary Row Operations
1) cR_i — multiply the ith row by the constant c
2) R_i \leftrightarrow R_j — swap the ith and the jth row
3) R_i+cR_j — add c times of the jth row to the ith row.

Gaussian Elimination Summary
Gaussian Elimination is essentially using the elementary row operations (in any order) to make the matrix to row echelon form.

Gauss-Jordan Elimination
After reaching row echelon form, continue to use elementary row operations to make the matrix to reduced row echelon form.

Vector Subspace Question (GRE 0568 Q3)

This is an interesting question on vector subspaces (a topic from linear algebra):

If V and W are 2-dimensional subspaces of \mathbb{R}^4, what are the possible dimensions of the subspace V\cap W?

(A) 1 only
(B) 2 only
(C) 0 and 1 only
(D) 0, 1, and 2 only
(E) 0, 1, 2, 3, and 4

To begin this question, we would need this theorem on the dimension of sum and intersection of subspaces (for finite dimensional subspaces):

\dim (M+N)=\dim M+\dim N-\dim (M\cap N)

Note that this looks familiar to the Inclusion-Exclusion principle, which is indeed used in the proof.

Hence, we have \dim(M\cap N)=\dim M+\dim N-\dim (M+N)=4-\dim (M+N).

\dim (M+N), the sum of the subspaces M and N, is at most 4, and at least 2.

Thus, \dim (M\cap N) can take the values of 0, 1, or 2.

Answer: Option D

If you are looking for a lighthearted introduction on linear algebra, do check out Linear Algebra For Dummies. Like all “For Dummies” book, it is not overly abstract, rather it presents Linear Algebra in a fun way that is accessible to anyone with just a high school math background. Linear Algebra is highly useful, and it is the tool that Larry Page and Sergey Brin used to make Google, one of the most successful companies on the planet.

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3×3 Matrix Inverse Shortcut (Fastest Method)


Backup copy (In case of broken link): 3×3 Matrix Inverse

This is an excellent method for finding the inverse of a 3×3 matrix, probably the fastest and easiest method for 3×3.

(Row reduction is better for 4×4 matrices and above.)

This method was first introduced to me by my student!

Featured book:

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Linear Algebra Resources


(How to diagonalize matrices)


(How to find exponential of matrices)


(Matrix Differential Equations)


(Jordan Normal Form)


(Jordan Normal Form for 2×2 Matrices)


(Matrix Exponential with repeated Eigenvalues)


Click to access matrixexp.pdf

(Matrix Differential equation with repeated Eigenvalues)


(Matrix Exponential of Block Diagonal)

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