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Types of Matrices



Square Matrices
The Identity Matrix
Diagonal Matrices
Triangular Matrices
Permutation Matrices
Symmetric Matrices
Skew-Symmetric Matrices
Orthogonal Matrices
Nilpotent and Idempotent Matrices
Involutory Matrices
Singular and Nonsingular Matrices
Summary of Matrix Types



Special Forms and Their Properties

Certain matrices satisfy a condition strong enough to determine their behavior in advance. The conditions fall into three kinds: a prescribed pattern of zeros, a relation to the transpose, or an equation in the powers of the matrix. Which kind a type belongs to predicts what can be said about its determinant, its eigenvalues, and its inverse — so the grouping is a tool rather than a filing scheme. Recognizing the pattern often turns a difficult computation into a straightforward one and settles which theorems apply.



Square Matrices

A matrix with equal numbers of rows and columns — nn rows and nn columns — is called square, and is said to have order nn. Square matrices occupy a privileged position in linear algebra because several fundamental concepts are defined exclusively for them.

Only square matrices have a determinant. Only square matrices can be invertible. Only square matrices have eigenvalues and a trace. Powers AkA^k are defined only when AA is square, since the product A⋅AA \cdot A requires the number of columns to equal the number of rows. Every type discussed on this page is a square matrix with additional structure imposed on top.

The Identity Matrix

The n×nn \times n identity matrix InI_n has ones on the main diagonal and zeros elsewhere:

Identity Matrix Definition
Iij=δij={1i=j0i≠jI_{ij} = \delta_{ij} = \begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases}

For example,

The Identity Matrix
I3=(100010001)I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

It is the multiplicative identity, leaving any compatible matrix unchanged:

Identity Matrix Property
AI=IA=AAI = IA = A

As a linear transformation, II is the map that sends every vector to itself.

The identity is simultaneously diagonal, symmetric, orthogonal, upper triangular, and lower triangular. Its determinant is 11, its inverse is itself, every eigenvalue is 11, its trace equals nn, and Ik=II^k = I for every non-negative integer kk. The subscript nn is dropped when the size is clear from context.
100010001
The identity, generated at this size

Ones the whole length of the diagonal, zeros everywhere else. Multiplying by this matrix leaves any compatible matrix untouched, and the picture shows why: each row of the identity selects exactly one row of whatever it meets and ignores the rest. Generate it at other sizes, alongside every other type on this page, on the matrix type generator.

Every notion of an inverse on this site is stated against this matrix: to invert something is to get back here.

Diagonal Matrices

A diagonal matrix has nonzero entries only on the main diagonal:

Diagonal Matrix Definition
D=diag(d1,d2,…,dn)=(d10⋯00d2⋯0⋮⋮⋱⋮00⋯dn)D = \text{diag}(d_1, d_2, \dots, d_n) = \begin{pmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{pmatrix}

Diagonal matrices are the easiest matrices to work with. Their arithmetic reduces to operations on the diagonal entries alone:

Diagonal Matrices
diag(d1,…,dn)⋅diag(e1,…,en)=diag(d1e1,…,dnen)\text{diag}(d_1, \dots, d_n) \cdot \text{diag}(e_1, \dots, e_n) = \text{diag}(d_1 e_1, \dots, d_n e_n)

Powers act entry-wise on the diagonal:

Diagonal Matrix Power
Dk=diag(d1k,…,dnk)D^k = \text{diag}(d_1^k, \dots, d_n^k)

Diagonal Matrices
D−1=diag(1/d1,…,1/dn)D^{-1} = \text{diag}(1/d_1, \dots, 1/d_n)

The inverse exists if and only if every diagonal entry is nonzero. The determinant is the product of the diagonal entries:

Diagonal Matrix Determinant
det⁡(D)=d1d2⋯dn=∏i=1ndi\det(D) = d_1 d_2 \cdots d_n = \prod_{i=1}^{n} d_i

and the eigenvalues are the diagonal entries themselves. As a transformation, a diagonal matrix scales each coordinate axis independently — stretching along axes where ∣di∣>1|d_i| > 1 and compressing where ∣di∣<1|d_i| < 1.
400070002
Diagonal: only the diagonal is free

Every entry off the diagonal is zero, so the whole matrix is described by the n numbers running down it. That is why products, powers and inverses of diagonal matrices reduce to arithmetic on those numbers one at a time. Set the diagonal entries yourself on the matrix type generator.

A diagonal matrix acts on each coordinate separately, stretching it by its own diagonal entry and leaving the others alone.

Triangular Matrices

An upper triangular matrix has all entries below the main diagonal equal to zero:

Triangular Matrices
U=(u11u12u130u22u2300u33)U = \begin{pmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{pmatrix}

A lower triangular matrix has all entries above the main diagonal equal to zero:

Triangular Matrices
L=(l1100l21l220l31l32l33)L = \begin{pmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & l_{33} \end{pmatrix}

Triangular matrices share several convenient properties with diagonal matrices. The determinant is the product of the diagonal entries:

Triangular Matrix Determinant
det⁡(T)=∏i=1ntii\det(T) = \prod_{i=1}^{n} t_{ii}

The eigenvalues are the diagonal entries.

RULE
The product of two upper triangular matrices is upper triangular, and the same holds for lower triangular matrices.

The inverse of an invertible upper triangular matrix is also upper triangular.

These properties make triangular matrices the natural endpoint of Gaussian elimination. Row reduction converts a general matrix into upper triangular form, and the LU decomposition factors a matrix into lower and upper triangular components, reducing system-solving to two simple back-substitution passes.
385062009
Upper triangular: everything below the diagonal is zero

The zeros form a solid block beneath the diagonal, which is what makes this shape so cheap to work with: the determinant is just the diagonal product, and a system in this form can be solved by substituting upwards with no elimination left to do. Flip to the lower form and compare on the matrix type generator.

Elimination is, in these terms, the business of turning an arbitrary matrix into this shape.

Permutation Matrices

A permutation matrix is a square matrix with exactly one entry equal to 11 in each row and each column, and zeros everywhere else. Equivalently, it is the identity matrix with its rows rearranged:

Permutation Matrices
P=(010001100)P = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}

The condition is a pattern of zeros, like the diagonal and triangular conditions above — a permutation matrix can be recognised by sight, without computing anything. What separates it from those two is that the nonzero positions are not tied to the diagonal.

Every permutation σ\sigma of {1,…,n}\{1, \dots, n\} gives exactly one such matrix, and every such matrix comes from exactly one permutation. Writing ei\mathbf{e}_i for the standard basis vectors, PσP_\sigma is the matrix whose jj-th column is eσ(j)\mathbf{e}_{\sigma(j)}. There are therefore n!n! permutation matrices of order nn.

What multiplication does. Left-multiplication reorders rows, right-multiplication reorders columns:

Permutation Matrices
PA  =  rows of A permuted,AP  =  columns of A permutedP A \;=\; \text{rows of } A \text{ permuted}, \qquad A P \;=\; \text{columns of } A \text{ permuted}

Nothing is scaled and nothing is combined — entries are only moved. This is why permutation matrices cost no arithmetic in practice: a library applies one by reindexing, never by multiplying.

Properties. Permutation matrices are orthogonal, since the columns are distinct standard basis vectors and are therefore orthonormal:

Permutation Matrix Inverse
P−1=PTP^{-1} = P^{\mathsf{T}}

The determinant is the sign of the permutation — +1+1 when σ\sigma is even, −1-1 when it is odd — which is the same fact as the rule that each row swap flips the sign of a determinant.

Permutation Matrix Determinant
det⁡(Pσ)=sgn⁡(σ)\det(P_\sigma) = \operatorname{sgn}(\sigma)

The trace counts the fixed points of σ\sigma, since a 11 lands on the diagonal exactly where σ(i)=i\sigma(i) = i. The eigenvalues are roots of unity: a cycle of length kk contributes the kk-th roots of unity, so the whole spectrum is read off the cycle type.

Closure and the connection to row swaps. The product of two permutation matrices is a permutation matrix, the inverse of one is one, and the identity is one — so the n×nn \times n permutation matrices form a group under multiplication, a copy of the symmetric group SnS_n sitting inside the invertible matrices.

Inside that group, the matrices that swap exactly two rows are the row-swap elementary matrices of Gaussian elimination. Every permutation matrix is a product of them, because every permutation is a product of transpositions. The relationship runs one way only: row-swap elementary matrices are permutation matrices, but a permutation matrix is elementary only when it swaps a single pair. The other two kinds of elementary matrix — scaling a row, adding a multiple of one row to another — are not permutation matrices at all, since they change values rather than only positions.

Where it shows up. LU decomposition with partial pivoting cannot always factor AA directly, because a zero may sit where a pivot is needed. Recording the row swaps in a permutation matrix repairs this, and the factorisation is stated as

PA LU Partial Pivoting
PA=LUPA = LU
Learn more about this formula: PA LU Partial Pivoting →

/linear-algebra/formulas]@

The PP is exactly the bookkeeping of which rows were exchanged on the way to upper triangular form.

Nearby families. Relaxing the definition in different directions gives the matrices a permutation is usually compared against. A signed permutation matrix allows ±1\pm 1 in place of 11, which permutes the axes and may flip their direction — these are the symmetries of the cube. A monomial matrix allows any nonzero value, and factors as a permutation times a diagonal matrix. A doubly stochastic matrix keeps the row and column sums equal to 11 but drops the requirement that the entries be 00 or 11; Birkhoff's theorem says every such matrix is an average of permutation matrices, which makes the permutation matrices the corners of that set.
Permutation of (1,2,3) Cycle structure Transpositions det trace
(1,2,3) → (1,2,3) identity, three fixed points 0 +1 3
(1,2,3) → (2,1,3) one swap, one fixed point 1 −1 1
(1,2,3) → (1,3,2) one swap, one fixed point 1 −1 1
(1,2,3) → (3,2,1) one swap, one fixed point 1 −1 1
(1,2,3) → (2,3,1) 3-cycle, no fixed point 2 +1 0
(1,2,3) → (3,1,2) 3-cycle, no fixed point 2 +1 0
Reading the table downward shows why the determinant is called the sign of the permutation rather than a separate quantity to compute. Each transposition is one row swap, one row swap flips the sign, and the parity of the swap count is the same however the permutation is decomposed.

Symmetric Matrices

A square matrix is symmetric if it equals its own transpose, meaning aij=ajia_{ij} = a_{ji} for every pair of indices:

Symmetric Matrix Definition
A=ATA = A^T

The matrix is determined by its entries on and above the diagonal — everything below is a mirror image.

Symmetric matrices arise constantly in practice. Covariance matrices, Hessians in optimization, adjacency matrices of undirected graphs, and distance matrices are all symmetric.

RULE
Any product of the form ATAA^T A or AATAA^T is symmetric regardless of the shape of AA, since (ATA)T=AT(AT)T=ATA(A^T A)^T = A^T (A^T)^T = A^T A.

The spectral properties of real symmetric matrices are exceptionally clean. Every eigenvalue is real — no complex eigenvalues can appear. Eigenvectors corresponding to distinct eigenvalues are automatically orthogonal. And the spectral theorem guarantees that every real symmetric matrix can be diagonalized by an orthogonal matrix: A=QDQTA = Q D Q^T where QQ is orthogonal and DD is diagonal. This is a much stronger conclusion than ordinary diagonalizability, which requires only an invertible change-of-basis matrix.

A symmetric matrix is called positive definite if xTAx>0\mathbf{x}^T A \mathbf{x} > 0 for every nonzero vector x\mathbf{x}. Positive definiteness is equivalent to all eigenvalues being strictly positive, and it guarantees the existence of the Cholesky decomposition A=LLTA = LL^T.
472758286
Symmetric: entries mirrored across the diagonal

Each entry above the diagonal is matched by an equal entry below it, so the matrix is unchanged by reflection in that line — it equals its own transpose. This is the shape that guarantees real eigenvalues and an orthogonal set of eigenvectors later on. Compare it with the skew-symmetric case, where the mirrored entries carry opposite signs, on the matrix type generator.

Symmetry is the strongest structural gift a square matrix can have, and the spectral results later in the section are its payoff.

Skew-Symmetric Matrices

A square matrix is skew-symmetric if its transpose equals its negative, meaning aij=−ajia_{ij} = -a_{ji} for all i,ji, j:

Skew-Symmetric Matrix Definition
AT=−AA^T = -A

Setting i=ji = j forces aii=−aiia_{ii} = -a_{ii}, so every diagonal entry must be zero.

Every square matrix admits a unique decomposition into a symmetric part and a skew-symmetric part:

Symmetric Skew Decomposition
A=12(A+AT)+12(A−AT)A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)

The first term is symmetric, the second is skew-symmetric, and this splitting is unique.

THEOREM
The eigenvalues of a real skew-symmetric matrix are either zero or purely imaginary — they come in conjugate pairs ±bi\pm bi with real eigenvalues restricted to zero.

For matrices of odd order, the determinant is always zero: det⁡(A)=det⁡(AT)=det⁡(−A)=(−1)ndet⁡(A)\det(A) = \det(A^T) = \det(-A) = (-1)^n \det(A), and when nn is odd, this forces det⁡(A)=0\det(A) = 0. For even order, the determinant can be nonzero.

In R3\mathbb{R}^3, the cross product a×b\mathbf{a} \times \mathbf{b} can be written as [a]×b[\mathbf{a}]_\times \mathbf{b}, where [a]×[\mathbf{a}]_\times is the 3×33 \times 3 skew-symmetric matrix

Cross Product Skew Matrix
[a]×=(0−a3a2a30−a1−a2a10)[\mathbf{a}]_\times = \begin{pmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{pmatrix}

This reformulates the cross product as a matrix-vector multiplication.
072-708-2-80
Skew-symmetric: mirrored entries, opposite signs

Each entry above the diagonal is matched below it by its negative, and the diagonal itself is all zeros — an entry that must equal its own negative has nowhere else to go. Pair colours mark which entries mirror each other. Switch between symmetric and skew-symmetric on the matrix type generator.

Adding a symmetric and a skew-symmetric matrix can produce any square matrix, which is why the two types are studied as a pair.

Orthogonal Matrices

A square matrix QQ is orthogonal if its transpose equals its inverse:

Orthogonal Matrix Definition
QTQ=QQT=I  ⟺  Q−1=QTQ^T Q = QQ^T = I \iff Q^{-1} = Q^T

This means the columns of QQ form an orthonormal set: each column has unit length, and distinct columns are perpendicular. The same is true of the rows.

The determinant of an orthogonal matrix is ±1\pm 1, since 1=det⁡(I)=det⁡(QTQ)=det⁡(Q)21 = \det(I) = \det(Q^T Q) = \det(Q)^2:

Orthogonal Matrix Determinant
det⁡(Q)=±1\det(Q) = \pm 1

When det⁡(Q)=+1\det(Q) = +1, the matrix is a rotation. When det⁡(Q)=−1\det(Q) = -1, it involves a reflection.

RULE
The defining geometric property is that orthogonal matrices preserve lengths: ∥Qx∥=∥x∥\|Q\mathbf{x}\| = \|\mathbf{x}\| for every vector x\mathbf{x}.

They also preserve dot products (Qx⋅Qy=x⋅yQ\mathbf{x} \cdot Q\mathbf{y} = \mathbf{x} \cdot \mathbf{y}) and therefore angles between vectors. A transformation that preserves all distances and angles is called an isometry, and the orthogonal matrices are precisely the linear isometries.

Common examples include rotation matrices in R2\mathbb{R}^2 and R3\mathbb{R}^3, reflection matrices across any line or plane through the origin, and permutation matrices that reorder coordinates. The inverse of an orthogonal matrix is its transpose — making it the cheapest matrix inverse to compute.

Nilpotent and Idempotent Matrices

A square matrix AA is nilpotent if some positive power of it equals the zero matrix:

Nilpotent Matrix Definition
Ak=O for some integer k≥1A^k = O \text{ for some integer } k \geq 1

The smallest such kk is called the index of nilpotency.

THEOREM
Every eigenvalue of a nilpotent matrix is zero, which forces both the determinant and the trace to vanish.

Nilpotent matrices have a useful algebraic consequence: the matrix I−AI - A is always invertible, with inverse given by a finite geometric series:

Neumann Series Nilpotent
Ak=O  ⟹  (I−A)−1=I+A+A2+⋯+Ak−1A^k = O \implies (I - A)^{-1} = I + A + A^2 + \cdots + A^{k-1}

The series terminates because Ak=OA^k = O, so there is no convergence issue.

A square matrix AA is idempotent if applying the transformation twice is the same as applying it once:

Idempotent Matrix Definition
A2=AA^2 = A

The eigenvalues of an idempotent matrix can only be 00 or 11, since λ2=λ\lambda^2 = \lambda implies λ=0\lambda = 0 or λ=1\lambda = 1. A striking identity links the rank and the trace:

Idempotent Rank Trace
A2=A  ⟹  rank(A)=tr(A)A^2 = A \implies \text{rank}(A) = \text{tr}(A)

because the trace counts the eigenvalues equal to 11, which is the dimension of the image.

Geometrically, idempotent matrices are projections. They project Rn\mathbb{R}^n onto the column space of AA along the null space. If AA is also symmetric, the projection is orthogonal.
Attribute Nilpotent Idempotent
Defining condition Aᵏ = O for some integer k ≥ 1 A² = A
Eigenvalues all equal to 0 only 0 or 1
Determinant always 0 (singular) 0 or 1
Trace 0 equals rank(A)
Key identity (I − A)⁻¹ = I + A + A² + ⋯ + A^(k−1) rank(A) = tr(A)
Geometric meaning iterative collapse to the zero vector projection onto column space along null space

Involutory Matrices

A square matrix is involutory if applying it twice yields the identity — it is its own inverse:

Involutory Matrix Definition
A2=IA^2 = I

RULE
The eigenvalues of an involutory matrix must satisfy λ2=1\lambda^2 = 1, so they are restricted to +1+1 and −1-1.

Reflections are the prototypical example: reflecting twice across the same line or plane returns every vector to its starting point. This is the third of the power conditions, and it constrains the spectrum the same way the other two do — by handing the eigenvalues a polynomial equation they cannot escape.

The matrix (0110)\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} is involutory — it swaps the two coordinates and swapping twice restores the original. It is also a permutation matrix, which is worth noting because the two conditions are independent: this matrix satisfies both, but most permutation matrices are not involutory and most involutory matrices are not permutations.

More generally, any matrix of the form 2P−I2P - I, where PP is idempotent, is involutory. Expanding gives (2P−I)2=4P2−4P+I=4P−4P+I=I(2P - I)^2 = 4P^2 - 4P + I = 4P - 4P + I = I, using P2=PP^2 = P. Geometrically this converts a projection into the reflection across the same subspace: instead of landing on the subspace, the vector travels the same distance again and comes out the other side.

Singular and Nonsingular Matrices

The classification of a square matrix as singular or nonsingular is not a structural pattern like symmetry or triangularity — it is a behavioral property that depends on the values of the entries.

A singular matrix is one with zero determinant — equivalently, less than full rank:

Singular Matrix Definition
A singular  ⟺  det⁡(A)=0  ⟺  rank(A)<nA \text{ singular} \iff \det(A) = 0 \iff \text{rank}(A) < n

Its columns are linearly dependent, and the system Ax=bAx = \mathbf{b} fails to have a unique solution for every b\mathbf{b}. As a transformation, a singular matrix collapses at least one dimension — its image is a proper subspace of Rn\mathbb{R}^n.

A nonsingular (invertible) matrix has nonzero determinant, full rank, and linearly independent columns and rows. The system Ax=bAx = \mathbf{b} has exactly one solution for every right-hand side, and the inverse A−1A^{-1} exists.

Any matrix type can be singular or nonsingular depending on its entries. A diagonal matrix is singular if any diagonal entry is zero. A triangular matrix is singular if any diagonal entry is zero. An orthogonal matrix is never singular, since its determinant is ±1\pm 1. A nilpotent matrix (other than the zero matrix of order 11) is always singular, since all its eigenvalues are zero.

Summary of Matrix Types

The defining property of each type, together with its most important consequence, can be collected for quick reference.

Four types are defined by where the zeros sit. The identity matrix (Iij=δijI_{ij} = \delta_{ij}) is the multiplicative identity. Diagonal matrices (off-diagonal entries all zero) have trivially simple powers, products, and inverses. Permutation matrices (one 11 per row and column) reorder coordinates and are always orthogonal. Upper and lower triangular matrices (zeros below or above the diagonal) have eigenvalues visible on the diagonal.

Three are defined by a relation to the transpose. Symmetric matrices (A=ATA = A^T) have real eigenvalues and orthogonal eigenvectors. Skew-symmetric matrices (A=−ATA = -A^T) have zero diagonal and purely imaginary eigenvalues. Orthogonal matrices (QT=Q−1Q^T = Q^{-1}) preserve lengths and angles.

Three are defined by a power condition. Nilpotent matrices (Ak=OA^k = O) have all eigenvalues zero. Idempotent matrices (A2=AA^2 = A) are projections with rank=tr\text{rank} = \text{tr}. Involutory matrices (A2=IA^2 = I) are their own inverse.

These categories are not mutually exclusive, and the three axes cut across each other freely. The identity matrix is diagonal, symmetric, orthogonal, triangular, idempotent, involutory, and a permutation simultaneously. A 1×11 \times 1 zero matrix is diagonal, symmetric, skew-symmetric, triangular, nilpotent, and singular. Permutation matrices are all orthogonal, but only the ones built from disjoint swaps are involutory. Recognizing which types a given matrix belongs to is often the fastest route to understanding its behavior.
The eleven types below are grouped by what their defining condition constrains: a pattern of zeros, a relation to the transpose, or a power of the matrix. That grouping is not filing — it predicts the columns. Every type defined by a transpose condition has something to say about its eigenvalues being real or imaginary; every type defined by a power condition has integer eigenvalues, because the condition is a polynomial equation the eigenvalues must satisfy.
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Matrix · types

Eleven types and what each one forces

Each type is a condition on the entries, and the condition determines everything beside it. Grouped by what the condition constrains — the pattern of zeros, the transpose, or a power of the matrix.

1gaps
Defined by a pattern of zeros5
Identity§ 2Iij=1I_{ij} = 1 if i=ji = j, else 00
det11
eigenvaluesall 11
inverseII
key consequenceAI=IA=AAI = IA = A
The multiplicative identity, and the one matrix that commutes with everything of its size. Every type below is a generalisation of it in some direction.
Diagonal§ 3nonzero entries only on the main diagonal
det∏di\prod d_i
eigenvaluesthe did_i
inverse1/di1/d_i entrywise
key consequenceproducts and powers act entrywise
Everything is entrywise — which is the whole reason diagonalization is worth the effort of computing eigenvectors.
Permutation§ 12exactly one 11 per row and column
det±1\pm 1 — the sign of the permutation
eigenvaluesroots of unity
inversePTP^{\mathsf{T}}
key consequencereorders coordinates
A zero pattern, not a power condition — you recognise one by looking at it. Orthogonal, so the inverse is the transpose, and it is the PP of PA=LUPA = LU, where it records the row swaps made during pivoting.
Upper triangular§ 4all entries below the diagonal are 00
det∏aii\prod a_{ii}
eigenvaluesthe aiia_{ii}
inverseupper triangular
key consequenceclosed under products
The determinant and the spectrum are both read straight off the diagonal, which is what makes LU worth computing.
Lower triangular§ 4all entries above the diagonal are 00
det∏aii\prod a_{ii}
eigenvaluesthe aiia_{ii}
inverselower triangular
key consequenceclosed under products
The mirror image, and the LL of an LU factorization. The transpose of one is the other.
Defined by a transpose condition3
Symmetric§ 5A=ATA = A^{\mathsf{T}}
detnot stated
eigenvaluesall real
inversesymmetric
key consequenceA=QDQTA = QDQ^{\mathsf{T}}, QQ orthogonal
The spectral theorem case: not merely diagonalizable but orthogonally so, which means P−1P^{-1} is a transpose and nothing has to be inverted.
Skew-symmetric§ 6AT=−AA^{\mathsf{T}} = -A — forces a zero diagonal
det00 when nn is odd
eigenvalues00 or ±bi\pm bi in pairs
inverseskew-symmetric
key consequencerank is always even
Setting i=ji = j in the condition gives aii=−aiia_{ii} = -a_{ii}, so the diagonal vanishes. Odd order then forces det⁡=0\det = 0, since the eigenvalues pair off and one is left over at zero.
Orthogonal§ 7QTQ=IQ^{\mathsf{T}}Q = I — orthonormal columns
det±1\pm 1
eigenvalues∣λ∣=1|\lambda| = 1
inverseQTQ^{\mathsf{T}}
key consequencepreserves lengths and angles
The inverse is free — it is the transpose. This is what makes orthogonal factors numerically valuable: they cannot amplify error, since they change no lengths.
Defined by a power condition3
Nilpotent§ 8Ak=OA^k = O for some k≥1k \geq 1
det00
eigenvaluesall 00
inversenone exists
key consequence(I−A)−1=I+A+⋯+Ak−1(I - A)^{-1} = I + A + \cdots + A^{k-1}
Never diagonalizable unless A=OA = O: every eigenvalue is zero, so a diagonal form would be the zero matrix and hence so would AA.
Idempotent§ 8A2=AA^2 = A
det00 or 11
eigenvalues00 or 11 only
inverseonly when A=IA = I
key consequencerank⁡(A)=tr⁡(A)\operatorname{rank}(A) = \operatorname{tr}(A)
A projection. Rank equals trace because the trace sums the eigenvalues and they are all zero or one — so it counts the ones.
Involutory§ 9A2=IA^2 = I
det±1\pm 1
eigenvalues+1+1 or −1-1 only
inverseitself
key consequenceA−1=AA^{-1} = A
Its own inverse. Reflections are the geometric case, and the eigenvalue −1-1 is the direction being reflected across.
Read down a column rather than across a row and the families appear. Every type defined by a power condition has eigenvalues confined to the roots of its own defining polynomial — {0}\{0\}, {0,1}\{0,1\}, {±1}\{\pm 1\} — because the condition is an equation the eigenvalues must satisfy. Every type in the zero-pattern group can be recognised by sight alone. The groupings are not filing — they predict the columns.
Eleven types and what each one forces·/linear-algebra/matrix/typesLearn Math Class
The power conditions are worth reading as polynomial equations. A2=AA^2 = A says every eigenvalue satisfies λ2=λ\lambda^2 = \lambda, so λ∈{0,1}\lambda \in \{0, 1\}; A2=IA^2 = I says λ2=1\lambda^2 = 1, so λ=±1\lambda = \pm 1; Ak=OA^k = O says λk=0\lambda^k = 0, so every eigenvalue is zero. Three types, three constraints, and all of them fall out of substituting an eigenvector into the defining equation.
Two entries are worth contrasting directly. A nilpotent matrix has every eigenvalue zero and is never diagonalizable unless it is the zero matrix — the eigenvalues say nothing is happening while the matrix plainly does something, which is precisely what a defective matrix looks like. An idempotent matrix also has a constrained spectrum, but it is diagonalizable, and its rank equals its trace. Same kind of condition, opposite structural outcome.