What does it mean to diagonalize a matrix?

To diagonalize a square matrix A is to write it as A = P D P⁻¹, where D is a diagonal matrix and P is invertible. The columns of P are eigenvectors of A and the diagonal entries of D are the corresponding eigenvalues. In the basis of eigenvectors the transformation is a pure scaling along each axis, which is what the diagonal form expresses.

How do you diagonalize a matrix step by step?

Find the eigenvalues as the roots of det(A − λI). For each eigenvalue, row reduce A − λI and read one eigenvector per free column. If there are n independent eigenvectors in total, put them as the columns of P and the eigenvalues in the same order down the diagonal of D. Compute P⁻¹, and check that P⁻¹ A P equals D. If there are fewer than n eigenvectors the matrix is not diagonalizable.

When is a matrix not diagonalizable?

A real matrix fails to diagonalize over the reals in two situations. Either some repeated eigenvalue has an eigenspace smaller than its multiplicity, so there are fewer than n independent eigenvectors, which makes the matrix defective; the shear [[1,1],[0,1]] is the standard example. Or the eigenvalues are complex, as for a rotation, in which case a diagonalization exists over the complex numbers but not with real matrices.

How is diagonalization used to compute matrix powers?

If A = P D P⁻¹ then A^k = P D^k P⁻¹, because every inner P⁻¹P in the product cancels. A power of a diagonal matrix is found by raising each diagonal entry to the power, so any A^k costs only two matrix multiplications. The same idea defines other functions of a matrix, such as the matrix exponential, by applying the function to the diagonal entries.

What is special about diagonalizing a symmetric matrix?

A symmetric matrix is always diagonalizable, its eigenvalues are real, and its eigenvectors for different eigenvalues are perpendicular. Normalizing the eigenvectors to unit length makes P an orthogonal matrix Q whose inverse is its transpose, so A = Q D Qᵀ with no inverse to compute. This is the spectral decomposition, and it is the basis of principal component analysis and the classification of quadratic forms.

Matrix Diagonalization Visualizer | A = PDP⁻¹ Step by Step

Key Terms

Diagonalization — writing

A = PDP^{-1}

with

D

diagonal.

$P$ — the matrix whose columns are

n

independent eigenvectors of

A

; invertible because they are independent.

$D$ — the diagonal matrix of eigenvalues, in the same order as the columns of

P

.

Diagonalizable — a matrix that has

n

independent eigenvectors; equivalently, one whose geometric multiplicities all equal the algebraic ones.

Defective — a matrix with too few eigenvectors; it has no diagonal form, only a Jordan form.

Similar matrices —

A

and

P^{-1}AP

; they share eigenvalues, trace and determinant. Diagonalization is similarity to a diagonal matrix.

Eigenvector basis — the columns of

P

; in this basis the map is a pure scaling.

Spectral decomposition — the symmetric case

A = QDQ^T

, where

Q

is orthogonal, so the inverse is the transpose.

Power formula —

A^k = PD^kP^{-1}

, with

D^k

obtained by powering the diagonal entries.

Getting Started with the Visualizer

Choose a matrix and a power, then watch it factor into

PDP^{-1}

.

• Use the Preset pills for eight matrices: a classic

2 \times 2

, a symmetric one, a Markov chain, the Fibonacci matrix, a

3 \times 3

with distinct eigenvalues, a repeated eigenvalue that still diagonalizes, a defective matrix, and a rotation
• Use the Size stepper for

2 \times 2

or

3 \times 3

, and the k stepper to choose which power

A^k

the last scene computes
• Edit any entry directly, or press Shuffle for a random matrix with small integer eigenvalues
• Hover the ? icon for a reminder of what diagonalization is and when it works
• Press play or step manually; the step log on the right keeps every stage

The eigenvalues and eigenvectors are reported, not derived, since the eigenvalue visualizer shows that derivation. This tool picks up where it stops: assembling

P

and

D

, inverting

P

, checking the product, and using the factorization for a power. Defective and complex cases stop early with an explanation.

The Scenes in Order

The visualizer follows the textbook procedure exactly.

• Eigenvalues — the characteristic polynomial and its roots, with the trace as a check
• Eigenvectors — one per free column of

A - \lambda I

for each eigenvalue, shown as columns above their eigenvalues; if there are fewer than

n

, the run stops with a defective verdict
• Assemble —

P

from the eigenvectors,

D

from the eigenvalues, in matching order
• Invert —

P^{-1}

by row reducing

[P \mid I]

, with the determinant of

P

noted
• Factor —

A = P \cdot D \cdot P^{-1}

multiplied out and compared with

A

• Verify —

P^{-1} A P = D

, the same identity from the other side
• Power —

A^k = P D^k P^{-1}

for the chosen

k

, with

D^k

obtained by powering the diagonal
• Done — the factorization with a note on symmetry and on how

A^k

behaves as

k

grows

Complex eigenvalues stop the run after the first scene, since no real

P

exists.

The Eigenvector Matrix

The first thing the tool checks is whether there are enough eigenvectors. The frozen picture below shows the default preset's eigenvalues

5

and

2

with their eigenvectors

(1, 1)

and

(1, -2)

as columns, each above its eigenvalue.

Two eigenvectors for a

2 \times 2

matrix, independent since they are not parallel:

A

is diagonalizable.

Eigenvectors, frozen

The default preset: eigenvalues 5 and 2, eigenvectors (1, 1) and (1, −2) as the columns on the right, each above its eigenvalue. Two independent eigenvectors for a 2×2 matrix - diagonalizable.

Independence is automatic when the eigenvalues are distinct, and that is the common case. The count only becomes a question for repeated eigenvalues, where it is decided by how many free columns

A - \lambda I

has: as many as the multiplicity means a full set, fewer means defective.

The order of the columns is a free choice. The tool sorts eigenvalues in decreasing order, so the dominant one comes first, but any order works as long as

D

follows it.

Assembling P and D

The frozen picture below shows

P

and

D

for the default preset side by side: the eigenvectors as columns, the eigenvalues down the diagonal in the same order, zeros elsewhere.

Together they encode the equation

AP = PD

, which is nothing more than the two eigenvector equations written as one matrix identity.

P and D, frozen

P with the eigenvectors as columns, D with the eigenvalues down the diagonal in the same order. Together they say A P = P D, one eigenvector equation per column.

Reading

AP = PD

column by column is the whole proof. The

j

-th column of

AP

is

A\mathbf{v}_j

, and the

j

-th column of

PD

is

\lambda_j\mathbf{v}_j

, since multiplying on the right by a diagonal matrix scales columns. The two agree exactly because each

\mathbf{v}_j

is an eigenvector.

D

is where all the information about the transformation sits.

P

is just a change of coordinates; the same

A

with differently scaled eigenvectors has a different

P

but the same

D

.

Inverting P

To move from

AP = PD

to

A = PDP^{-1}

requires

P^{-1}

, which exists because the columns of

P

are independent. The frozen picture below shows the default preset's

P

and its inverse, with thirds in every entry because

\det P = -3

.

The tool row reduces

[P \mid I]

; for a

2 \times 2

the formula does the same in one line.

P⁻¹, frozen

det P = −3, so the inverse has thirds everywhere: P⁻¹ = [2/3, 1/3; 1/3, −1/3]. Found by row reducing [P | I], or by the 2×2 formula.

The fractions are typical, not a sign of trouble.

P

is chosen with integer columns for readability, so

P^{-1}

carries

\det P

in its denominators, and the products

PDP^{-1}

clear them again.

For a symmetric matrix the inverse is free: with the eigenvectors normalized to unit length,

P

is orthogonal and

P^{-1} = P^T

. The symmetric preset demonstrates this, and it is the reason the spectral decomposition is written

QDQ^T

rather than

QDQ^{-1}

.

The Factorization

With all three pieces in hand,

A = PDP^{-1}

. The frozen picture below shows the default preset's factorization in full,

A

on the left and the three factors on the right.

The caption multiplies them out and recovers

A

exactly, which is the check that the eigenvectors and the inverse are right.

A = P D P⁻¹, frozen

The full factorization. Multiplying the three factors on the right reproduces A exactly, which checks both the eigenvectors and the inverse.

The factorization is best read right to left as three moves applied to a vector.

P^{-1}\mathbf{x}

gives the coordinates of

\mathbf{x}

in the eigenvector basis;

D

multiplies each coordinate by its eigenvalue;

P

turns the result back into standard coordinates. Nothing else happens. That is what it means for

A

to be a scaling in disguise.

The tool also checks the identity from the other side,

P^{-1}AP = D

: conjugating

A

by

P

leaves a diagonal matrix, and every off-diagonal zero in that product is a confirmation that the corresponding column of

P

is a genuine eigenvector.

Powers Through the Diagonal

The reason to diagonalize is what it does to powers. The frozen picture below shows the default preset's

A^3 = PD^3P^{-1}

:

D^3

is just

125

and

8

on the diagonal, and two multiplications produce

A^3

.

Change the

k

stepper to see any other power; the eigenvalues make the growth rate visible before the product is even computed.

A³ = P D³ P⁻¹, frozen

D³ is just 125 and 8 on the diagonal. Two multiplications give A³ = [86, 39; 78, 47], the same as A·A·A - and its trace 133 = 125 + 8.

The algebra is one line:

A^k = (PDP^{-1})(PDP^{-1})\cdots(PDP^{-1})

, and every interior

P^{-1}P

cancels, leaving

PD^kP^{-1}

. Powering a diagonal matrix is powering its entries, so the cost is constant in

k

, while direct multiplication costs

k - 1

products.

More important than the saving is what the formula says. The entries of

A^k

are combinations of

\lambda_i^k

, so the largest

|\lambda_i|

takes over as

k

grows:

A^k

grows along its eigenvector when that modulus exceeds

1

, settles when it equals

1

, and vanishes when it is below

1

. The same formula with

k

replaced by a function gives

e^{A}

,

\sqrt{A}

and every other function of a matrix.

A Markov Chain Settling Down

The Markov preset has eigenvalues

1

and

0.4

. The frozen picture below shows its eighth power:

0.4^8

is nearly zero, so

D^8

has essentially one non-zero entry and

A^8

is nearly the steady-state matrix, with both columns close to

(5/6, 1/6)

.

Raise the

k

stepper in the tool to watch the convergence.

Markov chain, eighth power, frozen

Eigenvalues 1 and 0.4; 0.4⁸ ≈ 0.00066, so D⁸ is nearly diag(1, 0) and A⁸ is nearly the steady-state matrix with both columns close to (5/6, 1/6).

This is the general picture for stochastic matrices. The eigenvalue

1

always exists, its eigenvector is the stationary distribution, and every other eigenvalue has modulus at most

1

. As

k

grows the powers

\lambda_i^k

of the others die out and

A^k

converges to a rank-one matrix whose columns are the steady state. The second-largest modulus, here

0.4

, is the convergence rate.

The Fibonacci preset shows the opposite behaviour. Its dominant eigenvalue is

\varphi \approx 1.618

, so the entries of

A^k

, which are Fibonacci numbers, grow like

\varphi^k

; the diagonalization written out is Binet's formula.

When It Fails

The defective preset is the shear

\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

. Its only eigenvalue is

1

, twice, but

A - I

has a single free column and a single eigenvector,

(1, 0)

. The frozen picture below shows the run stopping there.

One eigenvector cannot fill a

2 \times 2

matrix

P

, so there is no diagonal form.

Defective, frozen

The shear: λ = 1 twice but only the single eigenvector (1, 0). One column cannot fill P, so the run stops - no diagonal form exists.

Geometrically the shear moves every vector off the

x

-axis sideways, so no second direction is preserved. No change of basis can turn that into a scaling. The nearest replacement is the Jordan form,

\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

itself, a diagonal of eigenvalues with a

1

above it recording the missing eigenvector.

The other way to fail is complex eigenvalues, shown by the rotation preset. There the eigenvectors exist but are complex, so

A = PDP^{-1}

holds over

\mathbb{C}

and not over

\mathbb{R}

; in the real plane the matrix is a rotation-scaling and stays that way.

Reading the Scene Player

Each scene shows the matrices for one step with the arithmetic in the caption.

• The columns of

P

are colour-coded, one colour per eigenvector, and the eigenvalue row above them uses the same order
• The diagonal of

D

is blue and its zeros are grey, to stress that only the diagonal carries information
•

P^{-1}

is secondary throughout; it usually carries fractions with

\det P

in the denominator
• In the factor and power scenes the left-hand matrix,

A

or

A^k

, is the target being reproduced
• In the verify scene the diagonal of the product is the target: the off-diagonal zeros are the check
• Eigenvectors are scaled to small integers whenever possible; the Fibonacci preset shows decimals because its eigenvalues are irrational

Choosing a Matrix

The eight presets each make a different point.

• $2 \times 2$ classic — eigenvalues

5

and

2

with eigenvectors

(1, 1)

and

(1, -2)

;

\det P = -3

, so

P^{-1}

has thirds
• Symmetric — eigenvalues

3

and

1

with perpendicular eigenvectors; normalizing the columns of

P

gives an orthogonal

Q

and

A = QDQ^T

• Markov chain — eigenvalues

1

and

0.4

; raise

k

and watch

A^k

settle toward the steady state, whose columns are multiples of the eigenvector for

1

• Fibonacci —

\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}

with eigenvalues

\varphi

and

-1/\varphi

;

A^k

contains Fibonacci numbers, and the diagonalization is Binet's formula
• $3 \times 3$ distinct — eigenvalues

11

,

2

,

1

; the same matrix as the eigenvalue tool's default
• Repeated but fine —

\lambda = 1

twice with a plane of eigenvectors; repeated eigenvalues do not by themselves prevent diagonalization
• Defective — the shear, with

\lambda = 1

twice but a single eigenvector; the run stops
• Rotation — complex eigenvalues

\pm i

; no real diagonalization

Shuffle builds

A

as

PDP^{-1}

with an integer unimodular

P

, so the eigenvalues are small integers and the run always completes.

What Diagonalization Is

Suppose

A

has

n

linearly independent eigenvectors

\mathbf{v}_1, \ldots, \mathbf{v}_n

with eigenvalues

\lambda_1, \ldots, \lambda_n

. Put the eigenvectors as the columns of

P

and the eigenvalues down the diagonal of

D

. Then

AP = A\begin{pmatrix} \mathbf{v}_1 & \cdots & \mathbf{v}_n \end{pmatrix} = \begin{pmatrix} \lambda_1\mathbf{v}_1 & \cdots & \lambda_n\mathbf{v}_n \end{pmatrix} = PD

column by column, because each column of

AP

is

A\mathbf{v}_j = \lambda_j\mathbf{v}_j

. Since the columns of

P

are independent,

P

is invertible, and

A = PDP^{-1}, \qquad D = P^{-1}AP

Read right to left,

A

becomes three simple moves:

P^{-1}

expresses a vector in eigenvector coordinates,

D

scales each coordinate by its eigenvalue, and

P

converts back. In the eigenvector basis the transformation is a pure scaling; that is what "diagonal" means geometrically.

The condition is exactly

n

independent eigenvectors. Distinct eigenvalues guarantee it. A repeated eigenvalue is fine as long as its eigenspace is as large as its multiplicity; when it is smaller the matrix is defective and no

P

exists. Complex eigenvalues of a real matrix give a complex

P

and

D

, which is a diagonalization over

\mathbb{C}

but not over

\mathbb{R}

.

The payoff is functions of

A

. Because

A^k = PD^kP^{-1}

and

D^k

is just the diagonal entries raised to

k

, every power costs two multiplications, and the same trick defines

e^{A}

,

\sqrt{A}

, and any polynomial in

A

. For the theory, see the diagonalization theory page and the eigenvalues overview.

Key Properties

Facts that follow from

A = PDP^{-1}

.

• Powers:

A^k = PD^kP^{-1}

; negative

k

works too when no eigenvalue is zero
• Polynomials and functions:

f(A) = Pf(D)P^{-1}

with

f

applied entry by entry to the diagonal
• Invariants:

\det A = \prod \lambda_i

and

\operatorname{tr} A = \sum \lambda_i

, read straight off

D

• Distinct eigenvalues always give a diagonalizable matrix
• Symmetric matrices are always diagonalizable, with an orthogonal

P

:

A = QDQ^T

• Non-uniqueness: scaling or reordering the columns of

P

, with the same reordering of

D

, gives another valid factorization
• Similarity:

A

and

D

are similar, so they share rank, eigenvalues, characteristic polynomial and minimal polynomial
• Long-run behaviour:

A^k

is dominated by the eigenvalue of largest modulus; it grows if that modulus exceeds

1

, settles if it equals

1

, and dies out if it is below

1

• Defective matrices have no diagonal form; the Jordan form is the nearest replacement

Why It Matters

Diagonalization is how eigenvalues get used.

• Recurrences: the Fibonacci matrix diagonalized is Binet's closed formula, and the same works for any linear recurrence
• Markov chains:

A^k

converges to the steady state because the eigenvalue

1

survives while the others decay; the rate is the second-largest modulus
• Systems of differential equations:

\dot{\mathbf{x}} = A\mathbf{x}

decouples into

n

scalar equations in the eigenvector basis, with solutions

e^{\lambda_i t}

• Matrix exponentials:

e^{At} = Pe^{Dt}P^{-1}

, the workhorse of linear dynamics and control
• Quadratic forms: for symmetric

A

,

\mathbf{x}^TA\mathbf{x}

becomes a sum of

\lambda_i y_i^2

in the eigenvector coordinates, which classifies conics and tests definiteness
• Principal component analysis: diagonalizing a covariance matrix rotates the data onto its axes of greatest variance
• Graph algorithms: counts of walks of length

k

are entries of

A^k

for the adjacency matrix

Worked Example

Take the default preset,

A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}

Eigenvalues.

\det(A - \lambda I) = \lambda^2 - 7\lambda + 10 = (\lambda - 5)(\lambda - 2)

, so

\lambda = 5, 2

. Check: they sum to the trace

7

and multiply to the determinant

10

.

Eigenvectors. For

\lambda = 5

,

A - 5I = \begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix}

gives

x_1 = x_2

:

\mathbf{v}_1 = (1, 1)

. For

\lambda = 2

,

A - 2I = \begin{pmatrix} 2 & 1 \\ 2 & 1 \end{pmatrix}

gives

x_2 = -2x_1

:

\mathbf{v}_2 = (1, -2)

.

Assemble.

P = \begin{pmatrix} 1 & 1 \\ 1 & -2 \end{pmatrix}

,

D = \begin{pmatrix} 5 & 0 \\ 0 & 2 \end{pmatrix}

.

Invert.

\det P = -3

, so

P^{-1} = \frac{1}{-3}\begin{pmatrix} -2 & -1 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 2/3 & 1/3 \\ 1/3 & -1/3 \end{pmatrix}

.

Check.

PD = \begin{pmatrix} 5 & 2 \\ 5 & -4 \end{pmatrix}

, and

PDP^{-1} = \begin{pmatrix} 10/3 + 2/3 & 5/3 - 2/3 \\ 10/3 - 4/3 & 5/3 + 4/3 \end{pmatrix} = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} = A

.

A cube.

D^3 = \begin{pmatrix} 125 & 0 \\ 0 & 8 \end{pmatrix}

, so

A^3 = PD^3P^{-1} = \begin{pmatrix} 125 & 8 \\ 125 & -16 \end{pmatrix}\begin{pmatrix} 2/3 & 1/3 \\ 1/3 & -1/3 \end{pmatrix} = \begin{pmatrix} 86 & 39 \\ 78 & 47 \end{pmatrix}

.

Direct multiplication agrees:

A^2 = \begin{pmatrix} 18 & 7 \\ 14 & 11 \end{pmatrix}

and

A^3 = A^2A = \begin{pmatrix} 86 & 39 \\ 78 & 47 \end{pmatrix}

. Note that

86 + 47 = 133 = 125 + 8

: the trace of

A^3

is the sum of the cubed eigenvalues.

Common Mistakes

A few mistakes recur.

• Mismatched order — column

j

of

P

must be the eigenvector for entry

j

of

D

; reorder one and you must reorder the other
• Writing $P^{-1}DP$ — the eigenvectors go in

P

and

A = PDP^{-1}

; the reversed form is what you get if

P

holds the eigenvectors of

A^T

, not of

A

• Transposing instead of inverting —

P^{-1} = P^T

only when

P

is orthogonal, which requires orthonormal eigenvectors, guaranteed only for symmetric

A

and only after normalizing
• Reading a repeated root as defective — a repeated eigenvalue is only a problem if its eigenspace is too small; the repeated preset diagonalizes fine
• Assuming every matrix diagonalizes — shears and other defective matrices do not, and real rotations do not over the reals
• Powering $A$ entry by entry —

A^k

is a matrix power; only

D^k

can be computed entry by entry, and that is the point of the factorization
• Forgetting the check —

P^{-1}AP

must come out diagonal; any off-diagonal entry means a column of

P

is wrong

Related Concepts

Eigenvalues and eigenvectors — the input to diagonalization; the eigenvalue tool derives them.

Matrix inverse —

P^{-1}

is computed here by the same row reduction of

[P \mid I]

.

Matrix powers — the main application; the power tool multiplies directly, this one uses the eigenvalues.

Similar matrices and change of basis —

D = P^{-1}AP

is

A

written in the eigenvector basis.

Spectral theorem — the symmetric case with orthogonal

P

.

Jordan form — what replaces

D

when the matrix is defective.

Singular value decomposition — the factorization that works for every matrix, square or not, using two orthogonal bases instead of one eigenvector basis.

Markov chains and recurrences — where powers of a fixed matrix, and hence diagonalization, do the work.

Matrix Diagonalization

Step explanations

Key Terms

Getting Started with the Visualizer

The Scenes in Order

The Eigenvector Matrix

Assembling P and D

Inverting P

The Factorization

Powers Through the Diagonal

A Markov Chain Settling Down

When It Fails

Reading the Scene Player

Choosing a Matrix

What Diagonalization Is

Key Properties

Why It Matters

Worked Example

Common Mistakes

Related Concepts