ConceptLinear Algebra used inMSIAI-written · not yet proofread

Invertible Matrix

Also Inverse Matrix · Nonsingular Matrix · Regular Matrix · Singular Matrix · Invertierbare Matrix · Reguläre Matrix

A square matrix A is invertible if there is a matrix A⁻¹ with A A⁻¹ = A⁻¹ A = I; this holds iff A has full rank.

Know these firstMatrix Rank

Properties#

ProofWhy rank

n

⇔ invertible

By rank–nullity, $\operatorname{rank}(A) = n \iff \dim\ker(A) = 0$ , so the map $x \mapsto Ax$ is injective. Also $\dim \operatorname{im}(A) = n$ , so $\operatorname{im}(A) = \mathbb{R}^n$ and the map is surjective. A bijective linear map has a linear inverse, and its matrix is $A^{-1}$ . Conversely, if $\operatorname{rank}(A) < n$ , some $x \neq 0$ has $Ax = 0$ . Then $A^{-1}Ax = x \neq 0 = A^{-1}0$ is a contradiction.

Products and transposes: $(AB)^{-1} = B^{-1} A^{-1}$ and $(A^\top)^{-1} = (A^{-1})^\top$ .
Positive definite ⇒ invertible: if $A \succ 0$ (positive definite), all eigenvalues are $> 0$ , so $0$ is not an eigenvalue. MSI: $\Phi^\top \Phi \succ 0 \iff \operatorname{rank}(\Phi) = d$ .
2×2 inverse: $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ for $ad - bc \neq 0$ .

Example#

Sources#

MML p. 100 (Thm. 4.1: invertible ⇔ $\det(A) \neq 0$ , $2 \times 2$ inverse (4.2)), MML p. 103 (Thm. 4.3: $\det(A) \neq 0$ ⇔ full rank)
Use in MSI: MSI script p. 30 ( $\Phi^\top\Phi \succ 0 \iff \operatorname{rank}(\Phi) = d$ ), MSI script p. 32 (ill-posed case: $\Phi^\top\Phi$ singular)
Exercise: MSI Sheet 00, task 1 (not published) (invertibility of a $3 \times 3$ system matrix), task 2 (not published) (invertibility of a Jacobian)

Connections

Prerequisites know first1

Matrix Rank

Mentioned in inline links10

Jacobian MatrixANA
Square case ( $m = n$ ): if $J(\bar x)$ is an invertible matrix and $f$ is continuously differentiable, then $f$ is locally invertible around $\bar x$ (inverse function theorem). Newton’s method for $f(x) = 0$ needs $J(x_k)$ invertible in ev… §
Multivariable DifferentiationANA
| MSI Sheet 00, task 2 | compute $J$ and check invertibility at $\bar x$ | Jacobian Matrix, Invertible Matrix | §
Determinant
See Invertible Matrix for the other equivalent conditions. Geometrically, $\det(A) = 0$ means the columns are linearly dependent, so they span a flat parallelepiped with volume $0$ . §
The sign tells whether $A$ keeps or flips orientation. Turning from $e_1$ to $e_2$ goes counterclockwise. If the turn from $a_1$ to $a_2$ does too, $\det(A) > 0$ and the letter F stays readable. If the turn is clockwise (here: the columns… §
Eigenvalue
Singular ⇔ eigenvalue 0: $\det(A) = \prod_i \lambda_i = 0$ iff some $\lambda_i = 0$ . So $A$ is invertible iff $0$ is not an eigenvalue, and then $A^{-1}$ has the eigenvalues $1/\lambda_i$ (from… §
Linear Map
bijective $\iff \operatorname{rank}(A) = n = m$ , i.e. $A$ is an invertible matrix §
Matrix Rank
Gram matrix: $\operatorname{rank}(A^\top A) = \operatorname{rank}(A)$ . So the $n \times n$ matrix $A^\top A$ (always PSD) is invertible iff $A$ has full column rank. MSI uses this for least squares: $(\Phi^\top \Phi)^{-1}$ exists iff… §
Positive Semidefinite Matrix
Positive definite ⇒ invertible: $B \succ 0$ has no zero eigenvalue, so it is an invertible matrix. A PSD matrix is invertible iff it is positive definite. §
System of Linear Equations
Square, invertible case: if $m = n$ and $A$ is an invertible matrix, the unique solution is $x = A^{-1} b$ . In practice it is computed by Gaussian elimination, not by forming $A^{-1}$ . §
Determinants and Eigenvalues
Singularity tests. For $A \in \mathbb{R}^{n \times n}$ these are all the same statement (Invertible Matrix, Rank and Invertibility): §
Rank and Invertibility
The rank counts the linearly independent columns of a matrix. Comparing it with the number of rows and columns tells you whether the matrix is invertible and how many solutions $Ax = b$ has. In MSI this decides whether a least squares prob… §
Invertible Matrix — the square, full-rank case §

Search in these mentions →