Examples of singular matrix in the following topics:
-
- The matrix $B$ is the inverse of the matrix $A$ if when multiplied together, $A\cdot B$ or $B\cdot A$ gives the identity matrix.
- The definition of an inverse matrix is based on the identity matrix $[I]$, and it has already been established that only square matrices have an associated identity matrix.
- When multiplying this mystery matrix by our original matrix, the result is $[I]$.
- If an inverse has been found, then a quick check to be sure it is correct is to multiply it by the original matrix and see if the identify matrix results:
- This is called a singular matrix.
-
- It is called the Singular Value Decomposition and works for any matrix whatsoever.
- Any matrix $A \in \mathbf{R}^{n \times m}$ can be factored as
- $\Lambda \in \mathbf{R}^{{n \times m}}$ is a rectangular matrix with the singular values on its main diagonal and zero elsewhere.
- A direct approach to the SVD, attributed to the physicist Lanczos, is to make a symmetric matrix out of the rectangular matrix $A$ as follows: Let
- This is the singular value decomposition.
-
- Osteoblasts, which do not divide, synthesize and secrete the collagen matrix and calcium salts.
- As the secreted matrix surrounding the osteoblast calcifies, the osteoblast becomes trapped within it.
- Osteocytes maintain the mineral concentration of the matrix via the secretion of enzymes.
- They are able to communicate with each other and receive nutrients via long cytoplasmic processes that extend through canaliculi (singular = canaliculus), channels within the bone matrix.
- When osteoblasts get trapped within the calcified matrix, their structure and function changes; they become osteocytes.
-
- Singular value decomposition (SVD) is one method of identifying the factors underlying two-mode (valued) data.
- To illustrate SVD, we have input a matrix of 23 major donors (those who gave a combined total of more than $1,000,000 to five or more campaigns) by 44 California ballot initiatives.
- Figure 17.6 shows the "singular values" extracted from the rectangular donor-by-initiative matrix using Tools>2-Mode Scaling>SVD.
- The "singular values" are analogous to "eigenvalues" in the more common factor and components scaling techniques.
- Two-mode scaling of California donors and initiatives by Single Value Decomposition: Singular values
-
- The Latin has two Numbers,—the Singular and Plural.
- The Singular denotes one object, the Plural, more than one.
-
- Let $\mathbf{v}_i$ denote the ith column of the matrix $V$ .
- (The same argument applies to $U$ of course. ) The outer product $\mathbf{v}_i \mathbf{v}_i ^T$ is an $m \times m$ matrix.
- It is easy to see that the action of this matrix on a vector is to project that vector onto the one-dimensional subspace spanned by $\mathbf{v}_i$ :
- On the other hand, if we only include the terms in the sum associated with the $r$ nonzero singular values, then we get a projection operator onto the non-null space (which is the row space).
-
- The matrix has a long history of application in solving linear equations.
- A matrix with m rows and n columns is called an m × n matrix or $m$-by-$n$ matrix, while m and n are called its dimensions.
- A matrix which has the same number of rows and columns is called a square matrix.
- In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
- Each element of a matrix is often denoted by a variable with two subscripts.
-
- If only one person is in the subject, it's singular.
- Collective nouns (which refer to a group of beings or things as a single unit) are singular, and so take singular verbs.
- They take a singular verb form.
- Amounts take singular verbs because they are treated as units, which are singular nouns.
- Sums and products take singular verbs in mathematical equations.
-
- The extracellular matrix allows cellular communication within tissues through conformational changes that induce chemical signals, which ultimately transform activities within the cell.
- This transport primarily uses the vascular tissues (xylem and phloem); however, there are also structural modifications called plasmodesmata (singular: plasmodesma) that facilitate direct communication in plant cells.
-
- The matrix structure organizes employees by function and output to capitalize on strengths and improve efficiency.
- The matrix structure groups employees by both function and product .
- Balanced or functional matrix: A project manager is assigned to oversee the project.
- Strong or project matrix: A project manager is primarily responsible for the project.
- Representing matrix organizations visually has challenged managers ever since the matrix management structure was invented.