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# the number of matrices diagonal entry 5 and other entrees will be zero then the matrix is

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### Mohammed

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get the number of matrices diagonal entry 5 and other entrees will be zero then the matrix is from screen.

## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is

{\displaystyle \left[{\begin{smallmatrix}3&0\\0&2\end{smallmatrix}}\right]}

, while an example of a 3×3 diagonal matrix is

{\displaystyle \left[{\begin{smallmatrix}6&0&0\\0&0&0\\0&0&0\end{smallmatrix}}\right]}

. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix.

A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.

## Contents

1 Definition

2 Vector-to-matrix diag operator

3 Matrix-to-vector diag operator

4 Scalar matrix 5 Vector operations 6 Matrix operations

7 Operator matrix in eigenbasis

8 Properties 9 Applications 10 Operator theory 11 See also 12 Notes 13 References 14 Sources

## Definition

As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix = (,) with columns and rows is diagonal if

{\displaystyle \forall i,j\in \{1,2,\ldots ,n\},i\neq j\implies d_{i,j}=0.}

However, the main diagonal entries are unrestricted.

The term may sometimes refer to a rectangular diagonal matrix, which is an -by- matrix with all the entries not of the form , being zero. For example:

{\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-3\\0&0&0\\\end{bmatrix}}}

or

{\displaystyle {\begin{bmatrix}1&0&0&0&0\\0&4&0&0&0\\0&0&-3&0&0\end{bmatrix}}}

More often, however, refers to square matrices, which can be specified explicitly as a square diagonal matrix. A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix.

The following matrix is square diagonal matrix:

{\displaystyle {\begin{bmatrix}1&0&0\\0&4&0\\0&0&-2\end{bmatrix}}}

If the entries are real numbers or complex numbers, then it is a normal matrix as well.

In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".

## Vector-to-matrix diag operator

A diagonal matrix {\displaystyle D}

can be constructed from a vector

{\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}&\dotsm &a_{n}\end{bmatrix}}^{\textsf {T}}}

using the

{\displaystyle \operatorname {diag} }

operator:

{\displaystyle D=\operatorname {diag} (a_{1},\dots ,a_{n})}

This may be written more compactly as

{\displaystyle D=\operatorname {diag} (\mathbf {a} )}

.

The same operator is also used to represent block diagonal matrices as

{\displaystyle A=\operatorname {diag} (A_{1},\dots ,A_{n})}

where each argument

{\displaystyle A_{i}}

is a matrix. The

{\displaystyle \operatorname {diag} }

operator may be written as:

{\displaystyle \operatorname {diag} (\mathbf {a} )=\left(\mathbf {a} \mathbf {1} ^{\textsf {T}}\right)\circ I}

where

{\displaystyle \circ }

{\displaystyle \mathbf {1} }

is a constant vector with elements 1.

## Matrix-to-vector diag operator

The inverse matrix-to-vector

{\displaystyle \operatorname {diag} }

operator is sometimes denoted by the identically named

{\displaystyle \operatorname {diag} (D)={\begin{bmatrix}a_{1}&\dotsm &a_{n}\end{bmatrix}}^{\textsf {T}}}

where the argument is now a matrix and the result is a vector of its diagonal entries.

The following property holds:

{\displaystyle \operatorname {diag} (AB)=\sum _{j}\left(A\circ B^{\textsf {T}}\right)_{ij}}

## Scalar matrix

This section may be confusing or unclear to readers. In particular, many sentences use incorrect, awkward grammar and should be reworded to make sense. Please help clarify the section. There might be a discussion about this on the talk page.

A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple of the identity matrix I. Its effect on a vector is scalar multiplication by . For example, a 3×3 scalar matrix has the form:

{\displaystyle {\begin{bmatrix}\lambda &0&0\\0&\lambda &0\\0&0&\lambda \end{bmatrix}}\equiv \lambda {\boldsymbol {I}}_{3}}

The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.[a] By contrast, over a field (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its centralizer is the set of diagonal matrices). That is because if a diagonal matrix

{\displaystyle D=\operatorname {diag} (a_{1},\dots ,a_{n})}

has

{\displaystyle a_{i}\neq a_{j},}

then given a matrix {\displaystyle M} with

{\displaystyle m_{ij}\neq 0,}

the

{\displaystyle (i,j)}

term of the products are:

{\displaystyle (DM)_{ij}=a_{i}m_{ij}}

and

{\displaystyle (MD)_{ij}=m_{ij}a_{j},}

स्रोत : en.wikipedia.org

## Matrices: Meaning, Types, Formula & Examples

Matrices: ✓ Meaning ✓ Formula ✓ Adding ✓ Multiplying ✓ Types ✓ Uses ✓ Algebra ✓ Examples ✓ StudySmarter Original

## Matrices

Have you ever wondered how the social media sites like Facebook or Linked-In keep track of relationship between its users? Suppose we have the following information of 5 users. Tom is connected with Leo and Radha, Leo is connected with Radha and Daniel, Daniel is connected with Radha and Tom, and Riya is not connected with any of them. Pssh! That's a lot to keep track of just for 5 users. Now let us arrange this information in the tabular form with 0 denoting no connection and 1 denoting the connection between two users.

This arrangement with rows and columns is what we call a matrix. Thus matrices provide a way of organising, storing and working with information. This also makes them useful in data collection like seismic surveys.

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## What is a matrix?

Matrix is a rectangular arrangement of elements. The horizontal arrangement is called rows and the vertical arrangement is called columns. Matrices are usually enclosed within square brackets [ ] or parenthesis ( ).

Now we arrange the numbers 1, 4, 7, 0, 2 and 5 into 3 rows and 2 columns as follows.

Here the numbers 1, 4, 7, 0, 2 and 5 are called the elements of the matrix.

We have the following information on the number of chocolates and cookies Reena and her two friends Reeta and Riya possess:

Reena has 10 chocolates and 15 cookies.

Reeta has 25 chocolates and 10 cookies.

Riya has 18 chocolates and 23 cookies.

We can arrange this information in a matrix as

In this matrix, the first column represents the number of chocolates possessed by Reena, Reeta and Riya respectively. The second column represents the number of cookies possessed by Reena, Reeta and Riya respectively.

### What is the dimensions/order of a matrix?

The dimension, or order, of a matrix gives information of the number of rows and columns in a matrix. The order of the matrix is given by the number of rows followed by "×" (read as "by") and then the number of columns in the matrix. Moreover, the total number of elements in a matrix is the product of these two numbers.

For a matrix with 3 rows and 2 columns its dimension or order is given as 3 × 2. It is read as "3 by 2". The first number denotes the number of rows and the second number denotes the number of columns the matrix contains. The total number of elements in this matrix would be 6.

The matrix has 3 rows and 4 columns and so we can write its dimension to be 3 × 4. And as you can see the total number of elements is 34 = 12.

Similarly the matrix has 4 rows and 2 columns. The dimension of this matrix is 4 × 2. And the total number of elements in the matrix is 42 = 8.

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### Matrix elements

The components of a matrix are called its elements or entries. We can describe a particular element in a matrix by giving its position in the row and column arrangement.

The 2 × 2 matrix A = has the elements 3, 4, 7 and 8. The element 3 can be referred by its position in the matrix. It is located in the first row and first column in the matrix. We can describe the element 3 by the ordered pair (1, 1) and since we are pointing at the matrix A we write A(1, 1). The element 4 is described by A(1, 2), meaning it is the element that is located in the first row and second column of matrix A.

In general, a matrix with order 4 × 4 can be represented as

The matrix has 4 rows and 4 columns. The total elements in the matrix is 44 = 16. The (1, 1) element of the matrix is a, (1, 2) corresponds to b, (1, 3) to c, and so on.

Consider the matrix A=. The matrix A has 3 rows and 3 columns. So its order is given as 33. It has a total of 9 entries. The ordered pair A(1, 2) points to the element 30 in the matrix. A(2, 3) is the element 52 in the matrix.

### When are two matrices equal?

If we wish to compare two matrices and see if they are equal, first and foremost they should have the same number of rows and columns. Then we should check if they have the same elements and if they are positioned in the same places in the arrangement.

We say two matrices denoted by and are equal and write A = B, if and only if:

they are of the same order, that is the number of rows in the matrices A and B is the same and the number of columns in the matrices A and B is also the same;

their corresponding elements are equal, i.e., a = j, b = k, c = l, d = m, and so on.

Suppose we have the two matrices A = and B =. These two matrices are comparable since the order of A and B is 2 × 2. Now to show they are equal we have to show a = e, b = f, c = g and d = h.

If we know , then we can say that the corresponding elements should be equal. We get the equalities g + 3 = 7, 5 = g + 1, 6 = h + 4 and h + 1 = 3. From this we can find the value of g = 4 and h = 2.

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### Types of matrices

स्रोत : www.studysmarter.co.uk

## Can the main diagonal of a diagonal matrix contain zero?

Answer (1 of 6): Yes. A matrix is just a collection of numbers, so nothing stops one of the numbers on the diagonal from being just a zero. If you know your matrix comes from a special situation, that is a different thing. For example, if you know the matrix is constructed by taking dot products...

Can the main diagonal of a diagonal matrix contain zero?

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Sort Joe Eaton

PhD in Applied Mathematics and Computational Science, The University of Texas at Austin (Graduated 2001)Author has 351 answers and 66.2K answer viewsAug 30

Yes. A matrix is just a collection of numbers, so nothing stops one of the numbers on the diagonal from being just a zero.

If you know your matrix comes from a special situation, that is a different thing. For example, if you know the matrix is constructed by taking dot products of orthogonal func...

Related questions

How is a zero matrix a diagonal matrix?

Is [0] a diagonal matrix?

What happens if the diagonal of a matrix is 0?

Is a null matrix also a diagonal matrix?

What are the steps to diagonalize the identity matrix?

Chaz Bowen

BSME in Mechanical Engineering, Washington State UniversityAuthor has 243 answers and 4.1M answer viewsAug 27

If you take A square matrix A A such that A ij =0 Aij=0 whenever i≠j i≠j

to be your formal definition, then the answer is “yes". This is a very common definition.

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Alexander Farrugia

Used matrices extensively in his PhD thesis.Author has 3.2K answers and 21.6M answer viewsAug 28

Yes, sure. The definition of a diagonal matrix, ironically, does not force the main diagonal entries to be anything specific; it only forces all the entries outside the main diagonal to be zeros. As long as that condition is satisfied, the main diagonal entries can be anything you want, including zeros. In fact, the zero square matrix is an example of a diagonal matrix.

Justin Rising

The set of n×n n×n

diagonal matrices over a field

F F are both an algebra

and a commutative ring

under the standard matrix operations. That only works because we allow the main diagonal entries to be zero; if we didn’t, they wouldn’t really be any kind of nice structure.

Related questions

How do we show that all the diagonal elements of a skew symmetric matrix are zero?

Is it essential that a zero matrix is always a diagonal matrix?

Can main diagonal entries be all zero in the upper/lower triangular matrix?

What does it called when the elements above the main diagonal of a matrix are zero?

What if the main diagonal of a square matrices have zero?

Asma Mushtaq

more than 10 years of teaching experience Author has 79 answers and 10.4K answer viewsAug 27

yes and when it does it means one of the eigen values is zero

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Alon Amit

30 years of Linear Algebra.Author has 7.6K answers and 121.9M answer views4y

Related

Can all or some diagonal elements of a diagonal matrix be equal to zero?

Yes, of course. A diagonal matrix is one where all the non-diagonal entries are zero. The diagonal entries can be anything they want to be and nobody, I mean nobody, can take that freedom away from them.

Alexandru Carausu

Former University Associate Professor at Universitatea Tehnica "Gh. Asachi" Iasi (1966–present)Author has 805 answers and 52.3K answer viewsAug 29

YES. If a diagonal matrix contains an entry a_ii = 0 then the matrix is singular :

| A | = Π_1^n a_jj = 0 .

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Related

Why is the output zero? I wanted to do a diagonal sum of the matrix and find the difference of both the diagonal using C.

You do not need two nested loops to compute sum1; since the matrix is square, you could have written

for(int i = 0; i < n; ++i)

sum1 += arr[i][i]

Note also the use of spaces to make the code actually human-readable.

There is no need to iterate over values that will never be used. There is nothing that requires that the indices on the array not be the same variable.

The second one is even worse. You only do the addition for [0][2], [1,1] and [2][0], so you could have written it without any loops

sum2 += arr[0][2]; sum2 += arr[1][1]; sum2 += arr Bending Moment

B.Tech in Civil Engg. from Delhi Technological University (Graduated 2020)3y

Related

Can main diagonal entries be all zero in the upper/lower triangular matrix?

Sure why not. But note the point that determinant of such a triangular matrix will be zero because all principle diagonal elements are zero.

The main thing to call a matrix to be upper triangular matrix is that it fulfills the condition that all the elements below principle diagonal must be zero (similary for lower triangular) . So the matrix you are talking about fulfills this condition and hence you can call those matrices as upper/lower triangular matrices even their leading diagonal elements are zero.

स्रोत : www.quora.com

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Mohammed 15 day ago

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